IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 19 (2007) 463101 (30pp)

doi:10.1088/0953-8984/19/46/463101

TOPICAL REVIEW

Mesoscale structures in microemulsions
S Komura
Department of Chemistry, Graduate School of Science and Engineering,Tokyo Metropolitan University, Tokyo 192-0397, Japan
E-mail: komura@tmu.ac.jp

Received 6 July 2007, in final form 6 September 2007
Published 23 October 2007
Online at stacks.iop.org/JPhysCM/19/463101
Abstract
Due to the existence of intermediate mesoscopic internal structures, soft matter
exhibits various fascinating non-linear and non-equilibrium phenomena. In
this review article, we focus on microemulsions consisting of water, oil,and surfactant from the viewpoint of soft matter physics. Microemulsions
exhibit a rich phase behavior as the composition and/or the temperature isvaried. In the middle phase, oil and water mix in the presence of surfactant
molecules to form a mesoscopic bicontinuous structure. To explain thecomplex behavior of microemulsions, it is useful to employ phenomenologicalapproaches such as the Ginzburg–Landau theory or the membrane theory. We
discuss the Ginzburg–Landau theory and also review the Teubner–Strey model,
the Gompper–Schick model, and the two-order-parameter model. Based onthese models, we discuss the structure of the middle phase and its wetting
transition. The membrane theory proposed by Helfrich is also useful for
describing the physical properties of microemulsions. Various structures in
microemulsions, such as droplets, bicontinuous and network structures, areproperly accounted for by the curvature elasticity model. We focus on the
Exxon model which clarifies the physical origin of the middle phase. Within
the phenomenological level of description, we review the dynamical aspectsof droplet and bicontinuous microemulsions. We also give an overview
of microemulsions found in multicomponent polymeric systems (polymericmicroemulsions). A discussion on recent applications of microemulsions
completes the review.

Contents
1. Introduction
2. Microemulsions
2.1. Interfacial tension
2.2. Phase behavior
2.3. Phase diagrams
0953-8984/07/463101+30$30.00 © 2007 IOP Publishing Ltd

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J. Phys.: Condens. Matter 19 (2007) 463101

3. Ginzburg–Landau theory
3.1. Teubner–Strey model
3.2. Structure of the middle phase
3.3. Gompper–Schick model
3.4. Wetting transition by the middle phase3.5. Two-order-parameter model
4. Membrane approach
4.1. Curvature elasticity energy
4.2. Emulsification failure
4.3. Persistence length
4.4. Challenges to the bicontinuous structure
4.5. Exxon model
4.6. Origin of the middle phase
4.7. Network structure
5. Dynamics
5.1. Dynamics of microemulsion droplets
5.2. Dynamics of bicontinuous structure
6. Polymeric microemulsions
7. Recent applications of microemulsions
8. Conclusion
Acknowledgments
References

Topical Review

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1. Introduction
When we try to mix oil and water, they separate into two phases by creating a horizontal
interface between them. This everyday phenomenon occurs because the contact area between
the oil and water is minimized due to the large interfacial tension acting between them (3–5×10−2 J m−2 ). However, once we add surfactant molecules to such a binary system, the natureof the solution changes dramatically. A surfactant molecule contains both hydrophilic andhydrophobic parts. At the oil/water interface, surfactant molecules tend to orient themselves
by pointing their hydrophilic part towards the water and hydrophobic part towards the oil. As
a result, the effective interfacial tension decreases considerably. Thermodynamically stable
solutions consisting of oil, water, and surfactant are called microemulsions. Microemulsions
are almost transparent or light blue liquids. In contrast, milk and margarine which scatter lightare called macroemulsions.
Microemulsions are an important subject of colloid science or surface science, and there
are an enormous number of studies on them. From the viewpoint of applications, surfactants
are used in the preparation of food, drugs, cosmetics and oil recovery systems. Detailed
explanations of the subject are available in [1–3] and other textbooks on colloid chemistry [4].
It is not the purpose of this review article to cover all topics on microemulsions. Instead, I
would like to give an overview of the physical and universal aspects of microemulsions.
Among the various physical approaches, phenomenological models are quite useful in soft
matter physics. In such models, appropriate approximations are employed depending on thetime and length scales of interest. Phenomenological approaches are universal on mesoscopic
to macroscopic length and timescales. My main purpose here is to provide a concise reviewof the historically important phenomenological theories that describe mesoscale structures in
microemulsions. Hence, I did not try to make a complete list of the references in this area,
although I do cover the latest developments in this field as much as possible. It should be noted,
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however, that phenomenological models explain only one relatively simple aspect of the richand complex behaviors of such systems. There are various time and length scales involved,
and these should be investigated with different experimental and theoretical approaches. By
integrating all the knowledge obtained from various methods, one can draw a whole picture of
microemulsions.
Before describing microemulsions, I should stress here the importance of mesoscale
structures in soft matter. (General overviews of soft matter are available in text books [5–7] orin review articles [8–10].) Because the constituent molecules are relatively large, a mesoscopic
internal structure spontaneously forms in typical soft matter such as polymers, liquid crystals,
surfactants, or colloids. These structures play an essential role in determining the physicalproperties of the material. Examples of mesoscopic length scales (10–1000 nm) are the Flory
radius of a polymer chain, the sizes of colloidal particles, bilayer vesicles, and microemulsion
droplets. The pore sizes in aerosols and the domain sizes in microphase separated block
copolymers also appear to be of the mesoscopic scale.
When the scale of the internal structure is L , the entropic modulus of the material istypically given by G ∼ kB T /L 3 , where kB is the Boltzmann constant and T the temperature.
(Note that G has dimensions of energy per unit volume.) For a typical solid, L corresponds tothe atomic length scale, and the order of G ranges from 1010 to 1011 Pa at room temperature.
On the other hand, for colloidal crystals, for example, particles are periodically arranged inspace, and the typical distance between them is much larger than the atomic size. Hence themodulus of a colloidal crystal can be very small, e.g. 1–100 Pa, or 1010 times smaller than themodulus of ordinary solids.
The existence of mesoscopic length scales in soft matter is also reflected in the dynamical
properties. If we regard the time needed to diffuse through the distance between the particles
τ as a typical timescale of the system, we have τ ∼ L 2 /D , where D is the particle diffusion
constant. For simple liquids τ is about 10−12 s, i.e. the particle motion relaxes immediately.
For colloidal crystals as well as other soft matter, τ ranges from 10−3 to 1 s, which is extremely
slow. Hence, the mesoscopic length scale in soft matter leads to dramatic structural changes
under weak external fields accompanied by a long relaxation time. This is why non-linear
and non-equilibrium properties are significant in soft matter. However, there are importantunresolved problems when dealing with the equilibrium properties of soft matter. Although
this review mainly concerns microemulsions in equilibrium, it does provide some discussions
on time-dependent phenomena.
In the next section, we discuss the basic properties of microemulsions, e.g. the interfacial
tension and the phase behavior. Section 3 describes various Ginzburg–Landau theories that
capture the essential features of microemulsions. Among these models, the Teubner–Streymodel and its extensions are discussed in some detail. In section 4, we review membrane theory,
which is based on the notion of curvature elasticity. After that, we introduce the Exxon model
which can describe the phase behavior of microemulsions. Section 5 describes the dynamics ofdroplet and bicontinuous microemulsions. Section 6 briefly explains microemulsions foundin polymeric systems, i.e. polymeric microemulsions. Finally, section 7 provides recent
applications of mesoscale structures in microemulsions.
2. Microemulsions
2.1. Interfacial tension
As mentioned in the previous section, a microemulsion is a thermodynamically stable ternary
solution made of water, oil, and surfactant. Hydrophilic and hydrophobic parts coexist in3

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a surfactant molecule. Adsorption of such molecules to an oil/water interface considerably
decreases its interfacial tension. Below we explain why adsorption of surfactants reduces theinterfacial tension.
For simplicity, consider an insoluble surfactant which dissolves neither in oil nor in water.
If we add such surfactant molecules to a mixture of oil and water, they adsorb at the oil/waterinterface, and the area per surfactant molecule  decreases. However, since the interfaces
cannot adsorb an infinite amount of surfactant,  will saturate at a certain value  ∗ . What
happens if we further increase the amount of surfactant ? In order to incorporate more surfactant
while keeping  ≈  ∗ , the oil and water form a complicated inter-connected structure so that
there is a gain in the interfacial area. The interface characterized by  ≈  ∗ is called thesaturated interface [11].
We can express such a saturated state in terms of the free energy [12]. The free energy per
surfactant molecule in the adsorbed monolayer f a () is written as

fa () = σow  + f s (),(1)

where σow is the oil/water interfacial tension, and f s () represents the interaction among thesurfactant molecules. The effective interfacial tension is given by∂ f a ()∂ f s ()= σow +
= σow − (),
(2)
∂
∂
where  = −∂ f s /∂ is the surface pressure. Since f a () is minimized when  =  ∗ forsaturated interfaces, we have σ ( =  ∗ ) = 0. In other words, the effective interfacial tension
vanishes for a saturated interface. Hence, the interfacial tension is zero in the ideal case.
σ () ≡

2.2. Phase behavior
By changing the temperature or the composition, microemulsions can be made to exhibit a richphase behavior. As depicted in figure 1, three different types of phase separation take place inmicroemulsions. Figure 1(a) shows an O/W (oil-in-water) microemulsion in which oil dropletsof mesoscopic size are dispersed in water and form a droplet phase. Section 4 discusses what
determines the size of the oil droplet. This droplet phase coexists with an excess oil phase
O (almost pure oil that cannot dissolve in water), and the whole system is in a two-phasecoexistence state. Such a coexistence state is called a Winsor I microemulsion. The interfacial
tension between the two phases (O/W and O) is roughly σom ∼ 10−4 J m−2 , which is two ordersof magnitude smaller than the oil/water interfacial tension σow . The opposite situation occurs
in the W/O (water-in-oil) microemulsion shown in figure 1(c). Here an inverted droplet phasecoexists with an excess water phase, resulting in another type of the two-phase state called a
Winsor II microemulsion.
When the hydrophilic and hydrophobic parts of the surfactant are of comparable size, or
when its affinities for oil and water are balanced, the solution separates into three coexisting
phases. Such a state is called a Winsor III microemulsion as shown figure 1(b). Theintermediate phase due to the gravitational effect that is between excess oil and excess water iscalled the middle phase. In the middle phase, almost equal amounts of oil and water mix on
the mesoscopic length scale through the creation of enormous numbers of interfaces saturated
with surfactant. (In some references, the middle phase is called the ‘microemulsion phase’. In
this article, we will use the term microemulsions to refer to any solution containing oil, water,and surfactant.) As the amount of surfactant increases, the middle phase expands so that thetwo excess phases vanish and a one-phase state appears. In this situation, phase separation
does not take place while the oil and water mix completely on the mesoscopic scale. If wefurther increase the surfactant content, the self-assembling nature of amphiphilic molecules
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Figure 1. Phase separation and structures of microemulsions. A surfactant molecule has ahydrophilic head group and a hydrophobic tail group. (a) Oil-in-water (O/W) droplet phasecoexisting with an excess oil (O) phase. The whole solution is in a two-phase coexistence state(Winsor I microemulsion). (b) Middle (M) phase coexisting with excess oil (O) and excess water(W) phases. A bicontinuous structure of oil and water domains separated by surfactant monolayers
appears in the middle phase. The whole solution is in a three-phase coexistence state (Winsor III
microemulsion). (c) Water-in-oil (W/O) droplet phase coexisting with an excess water (W) phase.
The whole solution is in a two-phase coexistence state (Winsor II microemulsion).

induces liquid crystalline phases such as the lamellar phase, hexagonal phase, or cubic phase.
All of these phases are characterized by spatially modulated ordered (rather than random)
structures. The lamellar and hexagonal phases are analogous to the smectic and columnar
phases in thermotropic liquid crystals [13].
2.3. Phase diagrams
The phase behavior of ternary systems can be summarized by using the Gibbs phase triangle.
Figure 2, taken from [14], is the Gibbs triangle of a ternary system consisting of water, octane,
and C10 E5 (non-ionic surfactant) at 44.6 ◦ C. In this phase diagram, ‘1’, ‘2’, and ‘3’ indicate
the one-phase, two-phase, and three-phase regions, respectively, and ‘Lα ’ denotes the lamellar
phase. In the two-phase region, the solution separates into two different compositions given by
the end points of the straight lines, which are called ‘tielines’. In the three-phase region, thesystem separates into three different compositions corresponding to the vertices of the shaded
triangle. There is also a coexistence region between the one-phase state and the lamellar phase.
The Gibbs triangle of figure 2 is for a particular temperature. Figure 3 schematicallyillustrates how the phase behavior evolves as the temperature increases. Here the dotted lines
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Topical Review

Figure 2. Gibbs triangle of a ternary microemulsion consisting of water, octane, and C10 E5 (nonionic surfactant) at 44.6 ◦ C. ‘1’, ‘2’, and ‘3’ indicate the one-phase, two-phase, and three-phaseregions, respectively, and ‘Lα ’ denotes the lamellar phase. Reprinted with permission from [14].

Figure 3. Temperature dependence of the Gibbs triangle for a ternary system consisting of oil,water, and surfactant. The temperature increases from (a) to (e). TL and TU correspond to thetemperatures at the critical end points. A three-phase coexistence occurs in the temperature range
TL < T < TU . The filled circles indicate the critical points.

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Figure 4. Phase prism of a ternary microemulsion consisting of oil (O), water (W), and surfactant(S). The vertical axis indicates temperature T . The shaded plane is a cut-through of the phase prism
at a water-to-oil ratio of 1:1. On this plane, the horizontal axis is the surfactant concentration.

Figure 5. Phase diagram of temperature T versus surfactant concentration φs for a 1:1 water-to-oil
ratio. This phase diagram corresponds to the shaded plane in figure 4. ‘1’, ‘2’, and ‘3’ indicate theone-phase, two-phase, and three-phase regions, respectively.

represent the tielines, the shaded triangle corresponds to the three-phase region, and the filled
circles are the critical points at which the two phases become identical (liquid crystalline phases
are omitted). From this sequence of Gibbs triangles, we see that the three-phase coexistence(as depicted in figure 1(b)) only occurs when TL < T < TU . At either T = TL or T = TU
corresponding to the critical end points, two of the three phases become identical so thatthe interface between them vanishes. For other temperatures, the solution is a two-phasecoexistence state (figure 1(a) or (c)). By stacking a sequence of the Gibbs triangles along
the temperature axis, one can construct a phase prism like the one shown in figure 4.
The shaded plane in figure 4 shows a cut-through of the phase prism at a 1:1 ratioof water to oil. The vertical and horizontal axes correspond to temperature and surfactant
volume fraction, respectively. On this plane, the phase behavior is typically as in figure 5,
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Figure 6. Freeze-fracture microscopy image of a microemulsion in the one-phase region (the middle
phase). The scale bar indicates 200 nm. The structure is a bicontinuous one in which oil and water
regions are randomly inter-connected. Reprinted with permission from [18].

which is called a ‘fish’ diagram according to the shape of the three-phase body. In thisrepresentation, one can easily see that the three-phase coexistence occurs only when TL <T < TU . A temperature-induced transition from an O/W microemulsion (figure 1(a)) to a W/O
microemulsion (figure 1(c)) is called a phase inversion.
As stated before, oil and water mix in a single phase when the amount of surfactant isrelatively large. The largest surfactant composition that forms the three-phase body in figure 5
is denoted as φs∗ . In other words, φs∗ is the minimum amount of surfactant required to mix oil
and water in the one-phase state. In the applications of microemulsions, it is important to makeφs∗ as small as possible.
So far we have explained the phase behaviors and the phase diagrams of ternary
microemulsions. In the following, we describe several phenomenological models that explainthese interesting properties of microemulsions. Depending on the length scale, there are various
approaches such as lattice spin models, Ginzburg–Landau models, and membrane theories.
Here we shall mainly discuss the latter two approaches. Readers may find more complete
descriptions in [15–17].
3. Ginzburg–Landau theory
3.1. Teubner–Strey model
Figure 6 shows a typical one-phase microemulsion structure (the middle phase) obtained fromfreeze-fracture microscopy [18]. We see that almost equal amounts of oil and water mix toform a sponge-like bicontinuous structure. The typical size of this bicontinuous structure isabout 10–100 nm, which corresponds to the mesoscopic length scale. Why such a randombicontinuous structure is thermodynamically stable is an interesting question.
Small-angle x-ray or neutron scattering methods are useful to investigate such bicontinuous
structures. Although microemulsions do not scatter light and look transparent, they do scatter
x-rays and neutrons. Teubner and Strey suggested that the water–water structure function of
the middle phase fits the following functional form [19, 20]:

S(q) =8

a S(0),
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where q is the wavevector, q is its magnitude, and c, g , and a are the phenomenologicalparameters. In the middle phase, they satisfy c > 0, a > 0, and g < 0, but we also allow
g to take on a positive value. The values of the parameters depend on the composition and/or
the temperature.
When g < 0, the above structure function shows a peak at a non-zero wavevector a 1/4
q ∗ = (−γ )1/2
,
(4)
c
√
where γ = g/ 4ac. Roughly speaking, this means the typical length scale of the structure in
the middle phase is about 2π/q ∗ . The fact that equation (3) decays as q −4 in the high-q limit
is in accordance with Porod’s law reflecting the existence of a large number of interfaces [21].
As the surfactant concentration increases and the water-to-oil ratio stays constant at 1:1, thepeak position shifts to higher q so that the structure becomes smaller and the peak intensity
decreases, meaning less contrast between oil and water [22].
Teubner and Strey proposed a free energy functional that reproduces the structure
function (3). They chose the difference in the local volume fractions of oil and water as theorder parameter ψ(r). The suggested Ginzburg–Landau type of free energy functional is [19]
FTS [ψ(r)] = dr [c(∇ 2 ψ)2 + g(∇ψ)2 + aψ 2 ].
(5)
Notice that this free energy is invariant under the transformation ψ → −ψ .
3.2. Structure of the middle phase
We will use the Teubner–Strey model to discuss the structure of the middle phase. The realspace water–water correlation function G(r) can be obtained from the Fourier transform ofequation (3). The correlation function G(r) gives the probability of finding water at r giventhat water exists at the origin. When −1  γ  1, it has the following form:
ξλ
2πr,
G(r) =
e−r/ξ sin(6)232π crλwhere the correlation length ξ is
 1/4
4c
1
ξ=
,
(7)
a
(1 + γ )1/2
and the characteristic wavelength λ is
 1/4
λ
4c
1
=
.
2π
a
(1 − γ )1/2

(8)Equation (6) is an oscillating function whose amplitude decays exponentially. This oscillationreflects the ability of surfactant to induce structural ordering between oil and water.
The dimensionless parameter γ measures the ability of the surfactant to form internal
structures. Qualitatively speaking, the surfactant is ‘stronger’ when γ is negative and its
absolute value is larger, whereas it is ‘weaker’ when γ becomes positive. If γ > 1, thecorrelation function becomes the Ornstein–Zernike form which decays monotonically, and no
structure exists in the liquid. When γ < −1, the most stable structure is the periodic lamellar
phase. Within the mean-field approximation in which we neglect thermal fluctuations, thesebehaviors can be summarized in the phase diagram shown in figure 7.
In figure 7, there are two distinct phases, i.e. disordered and lamellar. The second-ordertransition between these two phases occurs at γ = −1. The correlation function G(r) has
an oscillating component for −1  γ  1 (structured-disordered phase), whereas it decays
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Figure 7. Phase diagram of the Teubner–Strey model in terms of γ = g/ 4ac. DL at γ = 1 isthe disorder line, and LL at γ = 0 is the Lifshitz line. The boundary between the lamellar and thedisordered phases is at γ = −1. In the structured-disordered phase (−1 < γ < 1), the correlation
function has an oscillating component, whereas in the unstructured-disordered phase (γ > 1), it
decays monotonically. Inside the structured-disordered phase, the structure function has a peak at a
finite wavevector q ∗ > 0 for −1 < γ < 0, and at q ∗ = 0 for 0 < γ < 1.

monotonically for γ > 1 (unstructured-disordered phase). The line γ = 1 separating thesestructures is called the disorder line although it is not a phase boundary. The line γ = 0 (org = 0) is the Lifshitz line at which q ∗ in equation (4) vanishes. When 0  γ  1, G(r)oscillates in space, but the structure function S(q) does not show any peak. This means thesurfactant is relatively weak. A peak appears in S(q) when −1  γ  0, indicating a welldeveloped structure in the middle phase.
3.3. Gompper–Schick model
The free energy (5) describes the structure of the middle phase, but it can only deal with a
one-phase state. To consider a multi-phase coexistence, Gompper and Schick extended the
Teubner–Strey model in the following way [23].
First, the term aψ 2 in equation (5) is replaced so that three-phase coexistence is possible.
Unlike the middle phase, the excess water phase (ψ > 0) or the excess oil phase (ψ < 0) does
not have a mesoscale structure. Hence, these two excess phases have a positive g because thepeak of the structure function is at q = 0. In general, g can be a function of ψ . From these
considerations, Gompper and Schick proposed the following free energy functional [23]:
FGS [ψ(r)] = dr [c(∇ 2 ψ)2 + g(ψ)(∇ψ)2 + f (ψ)],(9)
where f (ψ) and g(ψ) are given, for example, by⎧2
ψ > ψ0⎨ ω2 (ψ − ψb )2
f (ψ) = ω0 ψ
|ψ| < ψ0⎩
ω2 (ψ + ψb )2
ψ < −ψ0
10(10)

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Figure 8. (a) The situation in which the middle phase wets the oil/water interface. (b) The situationin which the middle phase does not wet the oil/water interface and forms a lens. (c) Schematic
representation of the interaction energy V () between the oil/water interface. The dotted line is theresult of the mean-field approximation; V () is minimized at finite  and hence the middle phase
does not wet the interface. The solid line takes into account van der Waals interactions and/or
thermal fluctuations; V () is minimized when  → ∞. In such a case, a macroscopically thickmiddle phase wets the oil/water interface.

andg(ψ) =b0|ψ| < ψ0b2

|ψ| > ψ0 .

(11)Here, the constants√ are taken to be b0 < 0 and b2 > 0. The function f (ψ) is continuous whenψ0 = ψb /(1 + ω0 /ω2 ), and is minimized when ψ = 0 and ±ψb . Using the above model,
Gompper and Schick investigated interfacial profiles, elastic properties of interfaces, and phasebehaviors of microemulsions. We refer the reader to [15] for their results.
3.4. Wetting transition by the middle phase
Here we briefly describe the wetting of the oil/water interface by the middle phase. This is
an interesting phenomenon in which the structure of the middle phase affects the macroscopic
behavior of the solution.
In section 2.2, we saw that the middle phase is located between the two excess phases.
This means the oil/water interface is wet by the middle phase. Such a wetting phenomenon
occurs for relatively short and weak (large γ ) surfactants. For longer and stronger (smallγ ) surfactants, the middle phase does not wet the oil/water interface and forms a lens as
depicted in figure 8. The transition from the non-wet state to the wet state is called the wetting
transition [24].
To judge whether the oil/water interface is wet by the middle phase, we write the excessenergy due to the interface as a function of the thickness of the middle phase ;σ () = σwm + σom + V ().

(12)
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Here, σwm and σom are the water/middle phase and oil/middle phase interfacial tensions,
respectively, and V () is the interaction potential between the interfaces. The effective oil/waterinterfacial tension σow is determined from the minimum of σ (). If the minimum occurs at finite
, the middle phase does not wet the oil/water interface. (A finite thickness is microscopic
and the middle phase cannot form a macroscopic phase.) On the other hand, if σ () takes aminimum value at infinite , then the interface will be wet by a macroscopically thick middle
phase.
The interfacial properties are strongly dependent on the bulk properties. Using their model,
Gompper and Schick showed that V () is similar to equation (6), which is the product ofan exponentially decaying function and an oscillating component [23]. A continuous wetting
transition is expected at γ = 1 since the minimum of V () continuously goes away to infinity
as γ → 1. (Notice that equation (8) diverges as γ → 1.) As shown in figure 8(c), however, theminimum of σ () always appears at finite  as long as −1 < γ < 1. One might conclude thatthe middle phase never wets the oil/water interface. Such a conclusion contradicts the observed
behavior in real microemulsions in which wetting indeed takes place.
To resolve this problem, we need to take into account the van der Waals interaction between
the interfaces and also the steric interaction due to thermal fluctuations. The van der Waals
interaction per unit area behaves roughly as W −2 , where W is the Hamaker constant. Because
the electron density of the middle phase takes an intermediate value between those of oil andwater, W −2 is always positive. Therefore, the minimum of V () at finite  is shifted upward,
as shown in figure 8(c), and the potential minimum goes away to  → ∞.
Thermal fluctuation plays a similar role as van der Waals repulsion. It causes two interfaces(water/middle phase and oil/middle phase) to fluctuate; they collide with each other so thatthe allowed configurations are restricted. This reduces entropy and leads to a steric repulsive
interaction. Thus, the same argument as above applies here. According to experiments using
various surfactants, the middle phase that wets the interface is characterized not only by γ < 1(below the disorder line) but also by γ < 0 (below the Lifshitz line) [25–27].
3.5. Two-order-parameter model
In the above Ginzburg–Landau model, the local volume fraction difference ψ(r) between oil
and water is chosen as the order parameter. According to the theory of critical phenomena,
the coefficient a in equation (5) is proportional to the temperature difference T − Tc measured
from the critical temperature Tc . Since a is taken to be positive in the above models, the middle
phase is expressed as a high-temperature disordered phase (ψ = 0). This means that within the
single-order-parameter model, the local structure (or the separation between oil and water) of
the middle phase is induced by thermal fluctuations.
A different approach is to introduce the surfactant concentration ρ(r) and construct a
two-order-parameter Ginzburg–Landau model using ψ and ρ [28, 29]. In this model, it isessential to consider the coupling between ψ and ρ . Since the free energy of the middle phaseshould be invariant under the transformation ψ → −ψ , the possible coupling terms are suchas ρψ 2 , ρ(∇ψ)2 , and (∇ 2 ρ)ψ 2 . For example, Komura and Kodama proposed the following
model [29]:FKK [ψ(r), ρ(r)] = dr [c(∇ 2 ψ)2 + g(∇ψ)2 − aψ 2 + uψ 4 + eρ 2 (ρ − ρs )2 − sρ(∇ψ)2 ],(13)
where the phenomenological parameters c, g , a , u , e, s , and ρs are all taken to be positive. The
last coupling term represents the fact that the surfactant adsorbs to the oil/water interface tolower its interfacial tension (represented by the coefficient of (∇ψ)2 ). For saturated interfaces,
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the parameters are chosen to satisfy g = sρs . Since the coefficient of the ψ 2 term is negativein equation (13), the middle phase is a low-temperature one. Using this model, Kodama andKomura investigated the phase separation dynamics, or the rheology of the middle phase [30].
A vector nature of the surfactant molecule was taken into account in a different two-orderparameter model proposed by Chen et al [31].
4. Membrane approach
4.1. Curvature elasticity energy
Microemulsions can be described from a different point of view from that of the Ginzburg–
Landau theory. As mentioned in section 2, surfactant molecules adsorb to oil/water interfaces
to form monolayers which are in the fluid state. One can regard such a monolayer as a twodimensional fluid whose thickness is negligibly small compared with its lateral size. Fluid
membranes are very flexible and their configurations are characterized by the membrane
curvature. Membrane theory is useful for describing large-scale phenomena such as the phasebehavior of microemulsions.
Helfrich proposed the following curvature elasticity energy per unit area of themembrane [32]:fc (c1 , c2 ) = 12 κ(c1 + c2 − 2c0 )2 + κ̄c1 c2= 2κ(H + c0 )2 + κ̄ K ,(14)

where c1 and c2 are the two principle curvatures. The combinations H = −(c1 + c2 )/2 and1
K = c1 c2 are the mean curvature and the Gaussian curvature, respectively. These quantities
are the two invariants of a two-dimensional surface. The coefficients κ and κ̄ are called thebending modulus and the saddle-splay modulus, respectively. Note that these moduli have thedimension of energy. While κ must always be positive because of the stability, κ̄ can be both
negative and positive. According to the Gauss–Bonnet theorem of differential geometry, theintegral of the Gaussian curvature K over the whole surface is constant as long as the topologyof the membrane is fixed.
The quantity c0 is the spontaneous curvature that reflects the asymmetry between the insideand outside of the membrane. As we explain later, the phase inversion mentioned in section 2.3
occurs when c0 changes sign. In the case of non-ionic surfactants, the spontaneous curvaturevanishes at the temperature where the three-phase body meets the one-phase region (seefigure 5). There are theories which relate the elastic moduli κ and κ̄ to the phenomenologicalparameters c, g , and a of the Teubner–Strey model in equation (5) [33, 34].
4.2. Emulsification failure
We now discuss the relation between the curvature elasticity energy and the phase behaviorof microemulsions [35]. Let the volume fractions of oil, water, and surfactant be φo , φw ,and φs , respectively. Due to the incompressibility condition, these quantities amount to
φo + φw + φs = 1. For simplicity, we assume that all surfactant molecules adsorb to oil/waterinterfaces to form monolayers of a thickness δ that is almost equal to the surfactant size.
Consider a droplet phase in O/W microemulsions. We see below that the droplet size isuniquely determined by the average composition. We denote the radius of the spherical oil
droplet by r , and the number of the oil droplets per unit volume by n . The following relations
1 In some references, the mean curvature is defined as H = +(c + c )/2, but according to differential geometry, it is12
more appropriate to define it with a negative sign.

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Figure 9. Phase diagram of various droplet structures based on the curvature elasticity model. The
horizontal axis is x = −κ̄/(2κ + κ̄) and the vertical axis is y = 3c̃0 δφo /φs (see the text). The
droplet phase coexists with the excess oil phase in the ‘emulsification failure’ region.

hold from the incompressibility condition; φo = (4π/3)r 3 n and φs = 4πr 2 δn . Hence, we have
3δφo
,
φs

φs3
.
(15)
36πδ 3 φo2
On the other hand, if we put c1 = c2 = c in equation (14) and minimize with respect to c,
we obtainκ̄ −1c̃0 = c0 1 +
.
(16)
2κ
r=

n=

Notice that r is given by the composition of oil and surfactant, while 1/c̃0 is the material
constant, and they generally differ. In the droplet phase, the competition between the twolength scales (r and 1/c̃0 ) determines the phase behavior.
When the amount of surfactant is large so that c̃0r < 1, the system is in a one-phase state
forming the droplet phase. In this situation, the droplet size decreases and the droplet number
increases as φs becomes larger. However, when φs is small enough so that c̃0r > 1, some ofthe oil cannot be incorporated in the droplets and exists as an excess phase. This results inmacroscopic phase separation, which is called emulsification failure. When this occurs, thedroplet size is always given by 1/c˜0 which minimizes equation (14). Emulsification failure
occurs when c̃0r ≈ 1.
So far, we have considered spherical droplets, yet they can be also cylindrical or flat(lamellar). The radius of a cylindrical droplet is given by 2δφo /φs . The (dimensionless)curvature energies per unit area of spherical ( f sph ), cylindrical ( fcyl ), and flat structures ( f lam )are respectively given by
f sph
1
2
= 2− ,(17)2yy2κ̃ c̃0

f cyl
9
3
,
=
(1 + x) −
2
2
16 y
2y
2κ̃ c̃0(18)

and f lam =0, where κ̃ = κ + κ̄/2, x = −κ̄/2κ̃ , and y = c̃0r . By comparing these energies, we
obtain the phase diagram shown in figure 9 [35].
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The cylindrical structure appears when the absolute value of the negative saddle-splay
modulus κ̄ is small, i.e. x < 1/3. As x becomes larger, only the spherical and lamellar
structures exist. The lamellar structure is more stable when c̃0 is smaller. When x = 0 (orκ̄ = 0), the energies of the spherical and cylindrical droplets are degenerate. This means thatspheres and cylinders coexist at x = 0.
The discussion up to now is correct when thermal fluctuations are negligible comparedwith the curvature energy. Since the entropic effect stabilizes small objects, the region of thespherical droplet expands when the entropy of mixing is taken into account. The effect ofthermal fluctuations is discussed in [36]. Moreover, the above theory has been generalized
to include translational entropy, cylinder length polydispersity, and radial polydispersity [37].
Recently, an analogy between emulsification failure and Bose–Einstein condensation has been
proposed [38, 39].
4.3. Persistence length
Various experiments indicate that the bending modulus of a monolayer is of the order of several
kB T . Hence, the membrane’s shape may be strongly affected by thermal fluctuations. Asa quantity characterizing the membrane undulation due to thermal fluctuations, the so-called
persistence length can be introduced [40], i.e.
4πκξκ ≈ a exp
(19)
.
αkB T
Here a is a microscopic length scale, and we assume α = 1 in the following discussion. On alength scale larger than ξκ , the correlation in the membrane normal direction is completely lost
and the membrane takes on a random configuration. On the other hand, the membrane can be
regarded as statistically flat on a small length scale.
For a length scale comparable to ξκ , the curvature energy competes with the configurational
entropy of the membrane. The effect of thermal fluctuations can be expressed in terms of a sizedependent effective bending modulus. For a membrane of size L , the renormalized bending
modulus is [17, 41, 42]2κeff (L) = κ 1 −kB Tlog(L/a) .
4πκ

(20)

This formula states that the large-scale bending modulus is reduced due to thermal fluctuations3 .
The effective bending modulus κeff vanishes when the membrane size L becomes comparable
to the persistence length ξκ . In this limit, there is no energy cost for bending.
Hereafter, we consider a ‘balanced’ microemulsion in which the amounts of oil and water
are almost the same (φo ≈ φw ) and the spontaneous curvature is close to zero (c0 ≈ 0). We
expect that the lamellar phase will appear in the absence of thermal fluctuations since a flat
membrane (c1 = c2 = 0) minimizes the curvature energy. In reality, however, a randombicontinuous structure appears instead of the lamellar phase when φs is relatively small (seefigures 5 and 6).
The existence of the bicontinuous structure can be qualitatively understood from the notionof persistence length [40]. In the lamellar phase, the distance between the monolayers dL isdetermined by the average composition dL = δφo /φs ≈ δ/2φs when φo ≈ φw ≈ 1/2
φs .
Following the argument for the droplet phase, the competition between the lamellar structure
2 The coefficients of k T /(4π κ) in [41] and [42] are different. However, this difference is not essential in theBfollowing discussion.
3 The saddle-splay modulus κ̄ is also affected by thermal fluctuations. The length ξ for which κ̄ vanishes correspondsκ̄
to the ‘topological’ persistence length [43–45].

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and the random bicontinuous structure is determined by the ratio ξκ /dL . The lamellar phaseis stabilized when ξκ > dL , whereas the random structure appears when ξκ < dL because themembranes fluctuate to gain entropy. This is why a random bicontinuous structure is observedat the tail of the ‘fish’ diagram (see figure 5).
Why does the three-phase body appear instead of the one-phase bicontinuous structure as
the surfactant amount decreases? This question is intimately related to the physical origin of
the middle phase. In the next subsections, we shall review some approaches to this problem.
4.4. Challenges to the bicontinuous structure
Talmon and Prager were the first to consider the bicontinuous structure of the middle phase [46].
In their picture, the whole space is divided into random Voronoi polyhedra, each filled witheither oil or water. All the surfactant molecules are assumed to adsorb to the flat interfaces
between oil and water. Their model takes into account the entropy of mixing between oil andwater. However, it does not consider the curvature energy of the membrane.
Jouffroy et al improved the Talmon–Prager model in the following way [12]. Instead ofdividing the space into Voronoi polyhedra, they considered cubic cells whose lattice constantis determined by the persistence length ξκ . They allow the area per surfactant  to fluctuate
around  ∗ . Furthermore, they considered the curvature energy by choosing the local radiusof curvature to be ξκ . This model successfully explains the two-phase coexistence but cannot
obtain the three-phase coexistence, because the cell size is fixed to the persistence length ξκ .
Widom proposed a different model in which the size of the cubic cell is variable and isdetermined in a self-consistent manner at each point on the phase diagram [47]. The surfactantmonolayers are assumed to form a compressible two-dimensional fluid in which the area
per surfactant  can vary significantly. However, this assumption is contradictory to what
happens at a saturated interface for which the area per surfactant is almost fixed to  =  ∗ .
Nevertheless, Widom could obtain the three-phase coexistence by introducing a microscopic
cutoff. He also concluded that the typical scale of the structure in the middle phase is of theorder of 10 nm. However, this length scale has nothing to do with the persistence length. Hence,
there was a problem in the Widom’s model whereby the variation of the phase diagram due tothe change in κ could not be reproduced.
4.5. Exxon model
Based on the above historical models, Safran and his group at Exxon Research and Engineering
proposed the following model [48–50]. As depicted in figure 10, they divide the space into
cubic cells whose size is d . Each cell is assumed to be occupied by either oil or water, and thesurfactant molecules adsorb to the interfaces. Half of the surfactant volume fraction is assigned
to oil and water volume fractions, i.e. φ = φw + φs /2.
The first important aspect of the Exxon model is that the cell size d is not determined by thepersistence length, but by the average composition of oil, water, and surfactant. The interface
is assumed to be saturated so that the area per surfactant is constant, i.e.  =  ∗ . Given thiscondition and using a random mixing approximation, the cell size can be determined asφ(1 − φ)d = 6δ
.
(21)
φs
The free energy (per unit volume) of mixing between oil and water is given bykB T
fmix = 3 [φ log φ + (1 − φ) log(1 − φ)].
(22)dNote that 1/d 3 is the number of cells per unit volume.
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Figure 10. Model of the middle phase with a random bicontinuous structure. The three-dimensional
space is divided into cubic cells of size d as given by equation (21). The shaded regions correspondto oil, the thick solid lines represent surfactant, and the rest is water.

Figure 11. Phase diagram obtained from the Exxon model. The horizontal axis is theoil concentration, and the vertical axis φ̃s = φs exp(1/t) represents the modified surfactant
concentration. The parameters are c0 = 0 and t = 0.2. The small φ region is magnified in theinset. The filled circle represents the critical point. Reprinted with permission from [48].

On the other hand, the curvature energy per unit volume is estimated as4
8πκeff (d)fcurv =
φ(1 − φ)[1 − c0 d(1 − 2φ)].
(23)
d3
The second important aspect of the Exxon model is that the effective bending modulusκeff (d) is used for the curvature energy. In terms of the persistence length ξκ , we can writeκeff (d) = −tκ log(d/ξκ ), where t = kB T /4πκ is the dimensionless temperature. The sum ofequations (22) and (23) is the total energy per unit volume.
A typical phase diagram for c0 = 0 and t = 0.2 is shown in figure 11. (The horizontal axis
is φ and the vertical axis is φ̃s = φs exp(1/t).) Here, all types of coexistence are reproduced inaccordance with the phase behavior described in section 2.3. The inset shows the tielines in thesmall φ region, and the filled circle indicates the critical point.
Figure 12 shows the phase diagram for equal volume fractions of oil and water (φo = φw ).
Here, the horizontal axis is surfactant concentration φ̃s , and the vertical axis is dimensionless
4 The factor 8π comes from the approximation that the membrane is regarded as a part of a sphere of radius d/2. The
product φ(1 − φ) is proportional to the total area of the oil/water interface.

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Figure 12. Phase diagram obtained from the Exxon model when the volume fractions of oil andwater are equal. The horizontal axis φ̃s = φs exp(1/t) is the surfactant concentration, and thevertical axis is the spontaneous curvature c0 scaled by the persistence length ξκ . The temperatureparameter is t = 0.151. Reprinted with permission from [50].

spontaneous curvature (the dimensionless temperature is t = 0.151) [50]. This phase diagram
is analogous to the ‘fish’ diagram in figure 5 if the spontaneous curvature is a smooth functionof temperature [51]. One can confirm that the spontaneous curvature vanishes when the threephase body meets the one-phase region.
4.6. Origin of the middle phase
Here, we briefly discuss why the Exxon model can describe the three-phase coexistence state.
Had we used a constant bending modulus κ instead of κeff (d) in equation (23), the totalenergy would scale as f mix + f curv ∼ φs3 since d ∼ φs−1 . This scaling means that there
is no characteristic length scale involved in the curvature energy [52]. We may conclude
that the phase transition never occurs, no matter how much the surfactant volume fraction
φs changes. However, the above contradiction disappears if equation (23) uses the effective
bending modulus κeff (d), because the persistence length enters as a new length scale throughκeff (d).
We mentioned before that the lamellar phase is stable when φs is large and ξκ > d ,while the random structure is stable for the opposite conditions. Even if φs is very small, theone-phase state should in principle persist as long as d increases according to equation (21).
As indicated by equation (22), however, the entropy dramatically decreases when d becomeslarge. Hence, the system prefers to be random at small length scales so that the excess oil andwater separate macroscopically. When d becomes comparable to the persistence length ξκ , itis energetically unfavorable for the membranes to fluctuate at a smaller length scale becauseof the curvature energy. As a result of the balance between curvature energy and entropy, thebicontinuous structure in the middle phase becomes random on the length scale ξκ . However,it should be stressed that this fact is a consequence of the model, not of the assumption. Basedon the Exxon model, the correlation and structure functions of microemulsions were calculated
in [53]. An extension of the Exxon model is given in [54, 55].
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Figure 13. Transmission electron microscopy picture of a microemulsion in which an oil region has
formed a network structure. Many Y-shaped junctions exist as indicated by the arrows. Reprinted
with permission from [61].

In section 2.3, we defined the minimum surfactant volume fraction φs∗ to mix oil and water.
Since d ≈ ξκ at the boundary between the three-phase body and the one-phase region, we put
φ ≈ 1/2 to obtain
3δ
φs∗ ≈
.
(24)
2ξκ
Hence, φs∗ is essentially inversely proportional to the persistence length ξκ .
After the Exxon model came out, Wennerström and Olsson proposed a simple free energy
for balanced microemulsions based on the flexible membrane model [56]. In their model, thefree energy per unit volume is given by

fWO = a3 φs3 + a5 φs5 ,(25)

where a3 can be negative and a5 is positive for the reason of stability. The first term comesfrom the length scale invariance of the curvature energy, as discussed before. The above model
can also explain three-phase coexistence when a3 < 0 [57]. This model and the Exxon model
were recently compared for balanced microemulsions [58].
4.7. Network structure
A bicontinuous structure appears when the spontaneous curvature is close to zero and the wholesystem is balanced. When the spontaneous curvature is large enough, we encounter droplet
phases in which each globule is separated (see section 4.2). However, according to recent
transmission electron microscopy experiments, the oil region forms a tubular network structure
when the oil concentration is small [59]. Figure 13 is a picture of a network structure madeof long tubes connected by Y-shaped junctions. To describe such a network structure, Tlusty
et al proposed a model in which the competition between ‘end caps’ and ‘junctions’ of thetubes plays an essential role [60–62]. In the following, we explain their model briefly. A more
extended treatment based on the spin model was given in [63].
First, we consider the network consisting of cylinders with different lengths. Let c(m) be
the number density of a cylinder of length m . (Here, the length of the cylinder is measured inunits of its diameter and hence m is dimensionless.) If the inside of the cylinder is filled withoil, the volume conservation condition is
φo = mc(m) dm.
(26)
Since the number density of a z -folded junction ρz is 1/z times the number density of an
isolated cylinder that has two end caps, it is given by

2
ρz =
(27)
c(m) dm.
z
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Here we fix z and consider the competition between the end caps of z = 1 and z -foldedjunctions.
By using these quantities, the free energy per unit volume can be written as [64]fn
= c(m)[log c(m) − 1] dm + ρz z + (1 − z)ρz log ρz ,
(28)
kB T
where z is the energy (divided by kB T ) required to form a single junction from z cylinders.
The first term of equation (28) is the translational entropy of the free cylinders, and the second
term corresponds to the curvature energy of the junctions. The last term represents the loss ofthe entropy when a single junction forms from z free end caps.
By minimizing the free energy (28) with respect to c(m) under the constraint of (27), we
obtainc(m) ∼ exp(−2z /z) ρz2(1−1/z) exp(−m/M),(29)
1−z/2 zφoeand thewhich is an exponentially decaying distribution function. Here, M ∼coefficient of order unity is omitted. Substituting equation (29) into (28) yields the minimized
free energy:fn= −ρz ∼ −φoz/2 e−z .
(30)
kB T
The above result has several interesting physical aspects. First, the junctions behave as
z/2
an ideal gas since the free energy has the form fn ∼ −kB Tρz . Second, the term −φo e−zreflects an effective attraction between the oil molecules due to the presence of the junctions.
3/2
In particular, the Y-shaped three-fold junction (z = 3) gives rise to a term −φo e−3 . This term
induces phase separation in the low φo limit by overcoming the repulsive excluded volumeinteraction term (proportional to +φo2 ). Accordingly, the phases of high and low density
junctions coexist in equilibrium. A detailed analysis indicates that the coexistence region forms
a closed loop [60].
Although a network structure forms when z  3, the junctions of z  4 seldom appear
because of the Boltzmann factor e−z . Hence it is enough to consider the competition between
the end caps of z = 1 and the junctions of z = 3. The transition between the cylindricalstructure and the network structure is determined by the condition ρ1 ρ3 or 3 − 1 log φo .
Tlusty et al numerically estimated the curvature energies of the Y-shaped junction and the end
cap to obtain the following relation [61, 62]:κκ̄
(1.42c0r − 1.04) − 4π,
3 − 1 = 4π
(31)
kB T
kB T
where r = 3δφo /φs is the radius of the spherical droplet5 .
As shown in figure 14, Tlusty et al were able to add a network structure to thephase diagram of figure 9 [61]. The network structure exists between the cylindrical andlamellar structures. Such a sequence of structural transitions is indeed observed in real
microemulsions [59].
5. Dynamics
5.1. Dynamics of microemulsion droplets
The bending modulus κ and the saddle-splay modulus κ̄ can be experimentally obtained fromvarious scattering experiments that can resolve fluctuations of oil/water interfaces. Consider a520

Since r differs by a factor of 2/3 compared with that in [61], 1.42 in equation (31) is replaced with 2.14 in [61].

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Figure 14. Phase diagram of a microemulsion obtained from the model of Tlusty et al. The
horizontal axis is 2c0 δφo /φs , and the vertical axis is 2δφo /(ξκ φs ). S is the sphere, C the cylinder, N
the network, and L the lamellar structure. S+C is the coexistence region between the spherical andcylindrical droplets. Reprinted with permission from [61].

spherical oil droplet of radius r0 . Let us denote the out-of-plane displacement of the membraneby a(θ, ϕ, t) depending on angles θ , ϕ and time t . It is convenient to expand a(θ, ϕ, t) in termsof spherical harmonics Ylm (θ, ϕ):a(θ, ϕ, t) =alm (t)Ylm (θ, ϕ).

(32)

l,m

Based on the curvature energy (14), Safran calculated the fluctuation amplitudes in thermal
equilibrium to be [65–67]
 −1
|alm |2 
3κ̄
3k B TkB T(l − 1)(l + 2) (l − 2)(l + 3) + 4c0r0 −
−
f (φ)
=
,
(33)
κ
κ
4πκ
r02
where

f (φ) =

1
[φ log φ + (1 − φ) log(1 − φ)]φ(34)is the entropy of mixing per droplet.
In the presence of hydrodynamic flow, the membrane restoring force balances with theviscous resistance force due to the surrounding fluid after a short initial period of motion. In
this case, the autocorrelation function of the out-of-plane displacement,alm (t)al  m  (0) = δll  δmm  |alm |2  exp(−t/τlm )(35)

decays exponentially with a relaxation time τlm . By employing the Stokes approximation forthe hydrodynamic equations, several people have calculated the relaxation time under various
conditions [66, 68, 69]. The most general form obtained by Seki and Komura is [70, 71]
13κ̄
3k B T
1
κ
−
f (φ)
,
= 3 (l − 2)(l + 3) + 4c0r0 −
τlm
κ
4πκ
Z (l)
ηr0

(36)

(l − 1)(2l 2 + 5l + 5)E + (l + 2)(2l 2 − l + 2).
(l − 1)l(l + 1)(l + 2)

(37)

with

Z (l) =21

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Here, E = η /η is the ratio of the viscosities of the interior and the continuous phase of thedroplets. A somewhat similar calculation was performed by Lisy et al [72].
The situation becomes simpler when the droplet phase is in equilibrium with the excess oilphase. Then the droplets achieve their optimum size r0 = 1/c̃0 , as given by equation (16). In
the presence of the entropic term, one generally has

kB T
κ̄
+
f (φ),(38)2κ
8πκ
instead of equation (16). By using this relation, we have the size polydispersity index p of the
microemulsion droplet,c0 r 0 = 1 +p2 =|a00 |kB T,=
28π(2κ + κ̄) + 2kB T f (φ)
4πr0

(39)

and the relaxation time τ2 for the ‘peanut-like’ deformation corresponding to the second-order
spherical harmonic (l = 2),
1
1
24
kB T
f (φ)
.
= 3 4κ − κ̄ −
τ24π
32 + 23 E
ηr0

(40)

The relaxation time can be measured by neutron spin-echo (NSE) spectroscopy [73, 74].
This technique measures the correlation functions of the Brownian motions in a sample. On
the other hand, the quantity p can be measured by small-angle neutron scattering (SANS) [75]or dynamic light scattering (DLS) [76, 77]. By using equations (39) and (40), the combinationof NSE and either SANS or DLS enables us to determine the two elastic moduli [78]. For
example, it was reported that κ = 0.92kB T and κ̄ = −0.38kB T for a system consisting ofoctane/water/C10 E5 [76]. Such a combined technique has recently been used to measure theeffect of pressure on the bending modulus [79, 80]. The dynamics of dense microemulsion
droplets forming ordered cellular phases have also been investigated by NSE [81].
5.2. Dynamics of bicontinuous structure
So far we have discussed the deformation of spherical microemulsion droplets. Here we
review some of the approaches to describe the dynamics of a bicontinuous structure. Milner
et al suggested the existence of a mode associated with the formation and destruction ofpassages between different parts of oil and water networks [82]. Such a topological relaxation
time is very long in phases with sharply defined interfaces and high activation energies.
Experimentally, a long-time diffusive relaxation mode was found in microemulsions consistingof SDS, octanol, octane, and brine [83].
Gompper and Hennes calculated the dynamic structure function of microemulsions byusing the time-dependent Ginzburg–Landau model [84, 85]. In the simplest case, the equationof motion for the order parameter ψ(r, t) (the difference between the oil and water volume
fractions) can be written as

∂ψ
δ FTS
= L∇ 2
,
∂t
δψ

(41)

where FTS is the Teubner–Strey free energy given by equation (5) and L is the Onsager
coefficient. Notice that ψ is a conserved quantity. In this case, the dynamic correlation functionisS(q, ω) =22

2 Lq 2.
ω2 + [2 Lq 2 (cq 4 + gq 2 + a)]2(42)

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Gompper and Hennes conducted a field-theoretic study with a coupling to a hydrodynamic
flow field, and showed that the correlation function oscillates in time [84]. Equation (41) was
generalized to the case of a large number of components by Marconi and Corberi [86].
Later, Nonomura and Ohta used a variational approach to derive the equations of motion
for microemulsions [87]. The hydrodynamic interaction acting in the bicontinuous structure
is taken into account through the Oseen tensor. They showed that the intermediate structure
function S(q, t) consists of two exponential functions:S(q, t)= (1 − f )e−11 (q)t + f e−22 (q)t .
S(q)

(43)

Here f is a small q -dependent constant, and the relaxation rate 11 contains the hydrodynamic
effect. NSE measurements on microemulsions containing C12 E5 [88, 89] and C10E4 [90]confirmed the validity of the above equation.
By extending equation (41), the rheology of bicontinuous microemulsions has been
investigated as well. In the presence of a flow v, the time evolution of the order parameter
ψ(r, t) can be written as

∂ψ
δ FTS
+ ∇ · (ψ v) = L∇ 2
,
(44)
∂t
δψ
where the velocity field is governed by the Navier–Stokes equation. Such a model was first
proposed by Pätzold and Dawson [91, 92]. Under a steady shear flow, Corberi et al predicted
a shear thinning behavior of microemulsions [93, 94]. Later Rapapa and Maliehe calculated
the dynamic structure function S(q, t) under the shear flow along the Lifshitz line [95]. They
predicted that S(q, t) shows multiscaling behavior, i.e. the characteristic length scales as t 7/6
in the flow direction, and as t 1/6 in directions perpendicular to the flow.
The above Ginzburg–Landau approach is valid for the intermediate and low-q regimes,
i.e. q  q ∗ where 2π/q ∗ is the typical domain size (see equation (4)). For the highq regime, q  q ∗ , Zilman and Granek developed a theory in which microemulsion istreated as an ensemble of independent membranes with random orientations [96, 97]. The
approximate expression for the intermediate structure function of an orientated membrane (inthe z -direction) is a stretched exponential function
S(q, t)≈ exp[−(qz t)β ],S(q)with β = 2/3 and the qz -dependent relaxation rate is
kB T 1/2 kB T 3q .
qz = 0.0246
κ
η z

(45)

(46)

To compare this equation with the experimental results, an average over all possible orientationsof the membranes should be performed [96, 97]. The Zilman–Granek theory has been
experimentally confirmed for microemulsions containing C12 E5 by light scattering [98] andNSE [89].
6. Polymeric microemulsions
This section discusses microemulsions that have been found in polymeric systems. In
these systems, diblock copolymers act as amphiphilic molecules. Various ordered structures
form in pure diblock copolymer melts as a result of the microphase separation [99].
Recently, attention has focused on polymer mixtures that contain diblock copolymers andhomopolymers. For example, in ternary mixtures consisting of AB block copolymer/A
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Figure 15. Transmission electron micrograph of a polymeric bicontinuous microemulsion
consisting of symmetric PE-PEP/PE/PEP mixtures. Reprinted with permission from [100].

homopolymer/B homopolymer, the AB block copolymer adsorbs to the interface between thetwo homopolymers.
Similar to low molecular weight surfactants, there is a loss in mixing entropy when block
copolymers adsorb to the interface. For diblock copolymer molecules, however, there is anadditional loss in stretching entropy since the adsorbed molecules become elongated at theinterface. Moreover, diblock copolymers spontaneously form various ordered structures bythemselves, which do not occur for small surfactants. Although there are several differences
between diblock copolymers and small surfactants, the mesoscale structures which are formedin polymeric ternary systems are analogous to those in ternary microemulsions consisting ofoil/water/surfactant.
Some years ago, Bates et al found a random bicontinuous structure between the lamellar
phase and the two-phase coexistence region in a polymeric ternary system [100–103].
The electron microscope picture in figure 15 is very similar to the bicontinuous structure
observed in the low molecular weight microemulsion in figure 6. Hence, they called thesestructures polymeric microemulsions, and these have been found in several polymer mixtures.
Concerning the physical origin of the polymeric microemulsions, Bates et al consideredthat the homopolymers swell the lamellar structure and thermal fluctuations destroy thelamellar order. There are theoretical predictions for efficient design of bicontinuous polymericmicroemulsions [104, 105].
Thermal fluctuations do not have much effect on polymeric systems because each polymer consists of a large number of monomers. Hence, the properties of polymeric systems canbe quantitatively described within a mean-field theory. In the self-consistent mean-field theory
originally formulated by Edwards [106], the many-body problem is reduced to solve the configuration of a single chain in an external field. However, since this theory has difficulty in dealing
with non-uniform systems, many approximations have been proposed over the past years. Recently, it became possible to solve the set of self-consistent equations numerically without using
any approximation [107]. This method has been applied to polymeric ternary mixtures by Matsen to estimate the elastic properties of a diblock copolymer monolayer and the effective interaction between monolayers [108–110]. Also using self-consistent field theory, Kodama et al investigated the physical origin of polymeric microemulsions. They suggested that polymeric microemulsions are not induced by thermal fluctuations, but appear as a result of microphase sepa24

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Figure 16. Bicontinuous structure calculated according to the real-space self-consistent field theory.
Black and white regions correspond to A and B homopolymer rich regions, respectively.

ration without any structural order [111, 112]. Figure 16 shows the structure of a polymeric microemulsion obtained from a real-space calculation. A different approach to incorporate the effect of thermal fluctuations in field theory of ternary blends was proposed by Düchs et al [113].
The dynamic properties of polymeric bicontinuous microemulsions have recently attracted
a lot of interest. For example, it was reported that shear flow transforms the microemulsion
phase into a three-phase coexistence at strong shears, and it is followed by macrophaseseparation of homopolymers [114, 115]. For an intermediate shear rate, the viscosity shows
shear thinning and the structure function is rendered anisotropic by the flow. The transientrheology [116] and linear viscoelasticity of polymeric microemulsions [117] have also been
investigated. In the latter work, experimental data were compared with the theoreticalpredictions by Pätzold and Dawson [91, 92]. Using Brownian dynamics molecular simulations,
Narayanan et al reproduced flow-induced phase transitions in ternary polymer blends [118].
Their results are in accord with the actual dynamical behavior of polymeric bicontinuous
microemulsions [114, 115].
7. Recent applications of microemulsions
Let us review some of the recent applications of mesoscale structures in microemulsions.
(i) An interesting situation occurs when a small amount of amphiphilic diblock copolymer isadded to a low molecular weight microemulsion. That is, there is a dramatic enhancementof the solubility between oil and water [119]. In figure 17, we show how the ‘fish’phase diagram is modified in the presence of block copolymers. The addition of theamphiphilic block copolymer shifts the one-phase region to a smaller amphiphile volume
fraction. Hence, the total volume fraction needed to form thermodynamically stable onephase microemulsions is lower if a diblock copolymer/surfactant mixture is used instead ofpure surfactant. Such a finding is very important and has been called the ‘boosting effect’.
Neutron scattering experiments demonstrated that the polymers form uniformly distributed
mushroom conformations on the surfactant membrane [120–123].
This solubility phenomena can be attributed to the increase in the effective bending
modulus caused by the addition of block copolymers. We should remind the reader that25

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Figure 17. Three-phase body (3) and adjacent one-phase region (1) for water/n -decane/C12 E5containing equal volume fractions of oil and water. Addition of the amphiphilic block copolymer
PEP10–PEO10 shifts the one-phase region to a smaller volume fraction φC+D . δv denotes thevolume fraction of polymer in the surfactant. Reprinted with permission from [120].

according to equation (24), the minimum surfactant volume fraction needed to mix oil andwater scales as φs∗ ∼ ξκ−1 , where ξκ is the persistence length defined by equation (19).
Hence, φs∗ becomes smaller for a larger bending modulus. In the mushroom regime, theeffect of polymer decoration on membrane elasticity is calculated to be [124]kB T π
σ (Rw2 + Ro2 ),κblock = κ +
1+
(47)
12
2
where σ is the number density of block copolymers within the membrane, and Rw/o isthe end-to-end distance of the hydrophilic/hydrophobic polymer block. The increase inbending modulus due to the polymers in the brush regime was calculated by Komura
and Safran [125]. Later, dynamics of bicontinuous microemulsions with and withoutamphiphilic block copolymers were compared using NSE and DLS [126].
(ii) In contrast to amphiphilic block copolymers, the addition of two homopolymer chains
diminishes the effect of the surfactant (‘inverse boosting effect’) [127]. In this case,a decrease in bending modulus was observed as homopolymer content increased. On
the other hand, homopolymers increase the viscosities of oil and water. Dynamicalmeasurements of the corresponding system revealed that the addition of homopolymer
indeed reduces the bending modulus [128, 129].
(iii) Recently, Nakaya et al investigated the effect of confinement of water-soluble polymers
inside spherical microemulsion droplets [130]. They showed that spherical droplets
deformed to prolate ellipsoid droplets upon confinement, while keeping the total surface
area and the volume of all the droplets constant. Increasing the degree of polymer
confinement even further caused an isotropic–nematic transition in the concentrated
droplet region. This result may be applicable to drug delivery.
(iv) It is known that systems composed of O/W microemulsion droplets linked by telechelicpolymers form a multiconnected transient network [131]. Here, telechelic polymers havea hydrophilic backbone with a hydrophobic group at each chain end. These mixtures
provide a wide range of technological applications such as cosmetics, paints, and oil
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recovery. In such systems, the large one-phase region consists of a fluid sol phase and a
viscoelastic gel phase separated by a percolation line. On the other hand, phase separation
into a dilute sol phase and a concentrated gel phase occurs in the two-phase region [131].
The phase behavior and structure of microemulsion–telechelic mixtures was theoreticallystudied by Zilman et al [132]. Here the phase transition is driven by the competition
between the translational entropy of the droplets and the configurational entropy of thepolymer connections between them. The immunity of such a transient network to random
degradation of the polymers has also been studied [133].
(v) Due to the presence of both polar and nonpolar solvents within the homogeneouscontinuum, microemulsions are often used as reaction media. For example, these systems
are used for preparative organic synthesis and in bio-organic reactions such as lipase
catalysed reactions. There have been many attempts to fix the mesoscale structures of the
microemulsion through chemical reactions within the system. For instance, polymerization
reactions can be used in microemulsions to produce analogs of latex particles [134].
Various other applications using reactions in microemulsions have been investigated inrelation to catalysis, drug delivery, etc. Chemical reactions in microemulsions weretheoretically studied by Ganesan and Fredrickson; they considered the effects of thermal
fluctuations on reaction kinetics [135]. Fluctuations increase the reaction rate but do notaffect the long-time dynamical evolution.
8. Conclusion
This review focused on microemulsions consisting of oil/water/surfactant and explained theirmesoscale behaviors from the viewpoint of soft matter physics. Several phenomenologicalapproaches turn out to be quite useful. For instance, the Ginzburg–Landau theories andmembrane theories are universal. We also discussed polymeric microemulsions as a related
system. It is quite surprising that the variety of mesoscale structures and macroscopic phasebehaviors of microemulsions can be described by fairly simple models.
We limited the discussion to cases where the amount of surfactant is relatively small.
Various liquid crystalline phases start to appear as the surfactant concentration increases.
Microemulsions exhibit various delicate properties because they have characteristics of both
liquids and liquid crystals. The structure, phase behavior, and dynamics of microemulsions are‘intermediate’ between those of liquids and liquid crystals.
Acknowledgments
This work is supported by KAKENHI (Grant-in-Aid for Scientific Research) on Priority Areas‘Soft Matter Physics’ and Grant No. 18540410 from the Ministry of Education, Culture, Sports,
Science and Technology of Japan.

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