THE JOURNAL OF CHEMICAL PHYSICS 131, 094508 共2009兲Morphological transition and emulsification failure in globular
microemulsions
N. Shimokawa1,a兲 and S. Komura2
1

Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Tokyo 192-0397, Japan

2

共Received 17 June 2009; accepted 3 August 2009; published online 4 September 2009兲
We consider the condensation transition of microemulsion droplets of oil which are dispersed inwater in the presence of surfactant. Since a macroscopic oil phase is formed due to this transition,
it is called “emulsification failure.” Based on the free energy approach, we determine the transitionlines between the spherical and the cylindrical droplet phases as well as the phase boundary lines ofthe emulsification failure. The phase diagrams are calculated by changing the physical properties ofthe surfactant monolayer such as the saddle-splay modulus and the spontaneous curvature. For anegative saddle-splay modulus, the spherical droplet phase coexists with the excess oil phase. In
some cases, a re-entrant transition 共sphere→ cylinder→ sphere兲 is expected to take place. For apositive saddle-splay modulus, the system undergoes a direct transition from the cylindrical droplet
phase to the macroscopically phase separated state. The sphere-to-cylinder transition line
approaches the emulsification failure boundary as the saddle-splay modulus becomes larger.
© 2009 American Institute of Physics. 关doi:10.1063/1.3212002兴
I. INTRODUCTION

Microemulsions composed of oil, water, and surfactant
are thermodynamically stable fluid mixtures in which surfactant molecules adsorb at oil-water interfaces to form
monolayers.1,2 Although such mixtures appear to be homogeneous on a macroscopic length scale, the internal structure
consisting of oil and water domains corresponds to a mesoscopic length scale. Hence microemulsions are considered to
be a typical example of structured fluids.3 Microemulsions
are not only interesting from the viewpoint of fundamental
colloid science but also useful for industrial applications
such as in the preparation of foods, drugs, cosmetic products,
or in the oil recovery systems. In biological systems, microemulsions are relevant to the transport of fat or to other solubilization phenomena.4
Various mesoscopic structures are known in microemulsions; the droplet phase where oil 共water兲 droplets are dispersed in a continuous water 共oil兲 region, or the bicontinuous
phase where both the oil and water domains form convoluted
structures in three dimensions. In the following, we consider
oil-in-water 共O/W兲 microemulsion droplet systems, but thesame results apply for inverted water-in-oil 共W/O兲 microemulsions as well. The shape of droplets can be either
spherical or cylindrical depending on the composition ortemperature. The structures of microemulsion droplets can bedirectly visualized such as using freeze fracture electron
microscopy5 or cryogenic temperature transition electron
microscopy.6
When either the amount of oil is too large or the amountof surfactant is too small, oil droplets can contain only alimited volume of oil so that the saturated droplet phase coa兲Electronic mail: shimokawa@chem.scphys.kyoto-u.ac.jp.

0021-9606/2009/131共9兲/094508/8/$25.00

exists with an excess oil phase.7–9 Such a phenomenon waslater called “emulsification failure” by Safran andTurkevich.10 As described below, they also suggested that,along the emulsification failure boundary, the droplet structure is determined by the curvature elasticity of surfactantmonolayers. Various experimental techniques such as
NMR self-diffusion and light scattering,11,12 viscosity
measurement,13 small angle neutron scattering,14 or small
angle x-ray scattering15 were used in order to identify the
droplet structure near the emulsification failure boundary. Allof these experiments showed that the droplets adopt a minimum size and correspond to a spherical shape. Since emulsification failure is usually unfavorable in various applications, it is important to predict the condition under which thismacroscopic phase separation takes place.
From the theoretical point of view, the complex phasebehavior of microemulsions has been successfully describedby the Helfrich’s curvature energy for flexible surfactant
films.16,17 In this model, the total free energy of a surfactantmonolayer is given by
f c = ␴A +

冕 冋
dA

册

1
␬共c1 + c2 − 2c0兲2 + ¯␬c1c2 ,2

共1兲where ␴ is the surface tension, A the area of the surfactant
film, ␬ the bending modulus, ¯␬ the saddle-splay modulus, c1
and c2 the two principle curvatures, and c0 the spontaneous
curvature. Based on this curvature elasticity model, the phase
diagrams of microemulsions showing various equilibrium
structures such as lamellae, spheres, cylinders, or networks
have been obtained.18–22 In the work by Safran et al.,18 theboundary line between the spherical droplet phase and theregion of emulsification failure was determined from thecondition under which the two length scales become equal.

131, 094508-1© 2009 American Institute of Physics

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094508-2

J. Chem. Phys. 131, 094508 共2009兲

N. Shimokawa and S. Komura

The first length scale is obtained by the minimization of theabove curvature energy while assuming a spherical droplet
shape, i.e., c1 = c2 = c. Neglecting the surface tension term,
this provides us with the typical length scale 1 / c̃0 = 共1 / c0兲⫻关共2␬ + ¯␬兲 / 2␬兴. On the other hand, the incompressibility
condition gives the second length scale r = 3a␾o / ␾s, where ais the surfactant size and ␾o and ␾s are the volume fractionsof oil and surfactant, respectively. When the amount of surfactant is large so that c̃0r ⬍ 1, the system is in a one-phasestate forming a spherical droplet phase. However, when ␾s issmall enough so that c̃0r ⬎ 1, a certain amount of oil cannot
be incorporated in the droplets and exists as an excess phase.
Hence the condition for the emulsification failure is given byc̃0r ⬇ 1.18
Although the above description captures the essential
feature of emulsification failure, it is not complete because
neither entropy nor size distribution of the droplets are taken
into account. In order to include these two effects, Sear andCuesta23,24 proposed a free energy formulation for sphericalmicroemulsion droplets. In addition to oil droplets, they also
considered the presence of micelles which do not contain any
oil. They determined the boundary of emulsification failure
according to the condition such that the total volume of oil
droplets diverges. Interestingly enough, they pointed out that
there is an analogy between emulsification failure and Bose–
Einstein condensation.23,24 This is because a macroscopic
condensed phase is formed in both phenomena even in theabsence of any interactions. It should be noted, however, thatthey did not consider the presence of cylindrical droplets, nor
did they discuss how the phase behavior depends on thesurfactant material parameters such as the saddle-splay modules or the spontaneous curvature. Another similar free energy formulation was employed in Ref. 25 which explains
the closed-loop coexistence regions observed in nonionicsurfactant microemulsions.
In this paper, we use the model by Sear and Cuesta inorder to study the emulsification failure of both spherical andcylindrical droplet phases. Such a study is necessary because
it was shown in some experiments that the excess oil phasecoexists not only with the spherical droplet phase but alsowith either the cylindrical droplet phase or the networkphase.6,15 In addition to the emulsification failure, we also
discuss the morphological transition between the sphericaland the cylindrical droplet phases. We describe in detail how
the phase transition lines depend on the surfactant parameters such as ¯␬ or c0. We will show that the sphere-tocylinder transition line approaches to the emulsification failure boundary as ¯␬ becomes larger. When ¯␬ is negative, thespherical droplet phase coexists with the excess oil phase.
We also expect that a re-entrant transition 共sphere→ cylinder→ sphere兲 may occur for some parameter regions.
When ¯␬ is positive and large enough, the system undergoes adirect transition from the cylindrical droplet phase to themacroscopically phase separated state.
In Sec. II, we first discuss the model describing theemulsification failure and the morphological transition in
globular microemulsions. In Sec. III, we present the resultsof the phase diagrams together with the free energies and thelr

end-cap cylindrical part
FIG. 1. Schematic representation of a cylindrical microemulsion droplet.
The two end caps consist of hemispheres of radius r, and the length of thecylindrical part is ᐉ. A spherical droplet corresponds to ᐉ = 0.

equilibrium distributions of microemulsion droplets. Some
discussions and comparisons to the other works are argued inSec. IV.
II. MODEL

We consider O/W microemulsions where oil droplets are
dispersed in water with surfactant monolayers at oil/waterinterfaces. We assume that the shape of oil droplets can beeither spherical or cylindrical. In addition to oil droplets, we
also consider the existence of spherical micelles which arecomposed of surfactant molecules only. The possibility offorming cylindrical micelles is excluded in the following.
In the following, all the quantities which have the dimension of length and energy are scaled by the surfactant
molecular size a and the thermal energy kBT, respectively
共kB is the Boltzmann constant and T the temperature兲. We
denote the dimensionless volume of oil droplet by v 共scaled
by a3兲 which can vary as v0 ⱕ v ⱕ ⬁. For a system with thetotal volume V, the dimensionless free energy per unit volume F / V 共measured in unit of kBT / a3兲 is given by
F/V =

冕

⬁

dv␳共v兲关log ␳共v兲 − 1 + f共v兲兴

v0+ ␳m共log ␳m − 1 + f m兲,

共2兲

where ␳共v兲 is the number density of the microemulsion droplet of volume v, f共v兲 is the internal energy of the microemulsion droplet, ␳m is the number density of the micelle, f m is
the internal energy of the micelle. The terms ␳共log ␳ − 1兲 and␳m共log ␳m − 1兲 are the mixing entropy terms of droplets andmicelles, respectively.
Next we need the expression of the internal energy f of adroplet coated by surfactant molecules. By modeling the surfactant monolayer as an elastic sheet, f can be given by theHelfrich’s curvature energy f c in Eq. 共1兲. Hereafter, we use
the notation such that ␴ 共scaled by kBT / a2兲, ␬ 共scaled bykBT兲, ¯␬ 共scaled by kBT兲, and c0 共scaled by 1 / a兲 are all dimensionless quantities. We will use the convention that thepositive curvature corresponds to the case when the surfactant sheet is curved toward oil. We consider a cylindricalmicroemulsion droplet which consists of a cylindrical partand two hemispheres as shown in Fig. 1. The radius and thelength of the cylindrical part are denoted by r and ᐉ 共both inunit of a兲, respectively. A spherical droplet corresponds tothe case when ᐉ = 0. Here the polydispersity in the length ofthe cylindrical part is not considered because it has only little
effect on the phase behavior.21 Then the internal energy of acylindrical droplet f in Eq. 共2兲 is given by

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094508-3

J. Chem. Phys. 131, 094508 共2009兲

Transitions in globular microemulsions

F/V = − ␭␰ + ␮␾ − ␥ − ␳m ,f共r兲 = 4␲共␴ + 2␬c20兲r2 + 2␲共␴ᐉ + 2␬c20ᐉ − 8␬c0兲r+ 4␲共2␬ + ¯␬ − ␬c0ᐉ兲 +

␲␬ᐉ
.
r

共3兲

Here the variable has been changed from v to r according to
the relation v = 共4␲r3 / 3兲 + ␲ᐉr2. Since the radius of a spherical micelle is taken to be equal to the surfactant size a, its
internal energy f m can be obtained by setting ᐉ = 0 and r = 1
in the above equation. Then we havef m = 4␲共␴ + 2␬c20兲 − 16␲␬c0 + 4␲共2␬ + ¯␬兲.

共4兲In microemulsions, there are two imposed constraints,
i.e., both the total amount of surfactant 共denoted by ␰兲 andthe total amount of oil 共denoted by ␾兲 should be conserved.
These conditions can be written as

␰=

冕

⬁

d v s ␳ 共 v 兲 + s m␳ m

共5兲

v0

and

␾=

冕

⬁

dvv␳共v兲.

共6兲

v0

In Eq. 共5兲, s = 4␲r2 + 2␲rᐉ and sm = 4␲ are the surface areasof a microemulsion droplet and a spherical micelle, respectively.
The equilibrium distribution of microemulsion droplets
can be obtained by the minimization of Eq. 共2兲 under theconstraints of Eqs. 共5兲 and 共6兲. This is equivalent to requiring␦
共F/V + ␭␰ − ␮␾兲 = 0,␦␳共7兲

where ␭ is the Lagrange multiplier and ␮ the chemical potential. Then we can obtain the equilibrium distribution of
microemulsion droplets as

␳ = exp共− f − ␭s + ␮v兲.

共8兲

Similarly, the equilibrium distribution of micelles is obtainedby minimizing Eq. 共2兲 with respect to ␳m under the constraints of Eqs. 共5兲 and 共6兲. Then we get␳m = exp共− f m − ␭sm兲.

共9兲

In order to discuss the condition for emulsification failure, we look at the dependence of ␾ on v 关see Eqs. 共6兲 and共8兲兴. In Eq. 共8兲, both f and s are irrelevant in the limit of v
→ ⬁ because they are proportional to r2. Hence the distribution of droplets asymptotically behaves as ␳ ⬃ e␮v whenv → ⬁. For ␮ ⱕ 0, the distribution ␳ approaches zero and ␾
takes a finite value. For ␮ ⬎ 0, on the other hand, ␳ diverges
so that ␾ becomes infinitely large. Notice that the case of␾ → ⬁ corresponds to the formation of a macroscopic oilphase, and hence the emulsification failure. Therefore thecondition for emulsification failure is set by ␮ = 0.
By substituting back the equilibrium distributions ofdroplets ␳ and micelles ␳m into Eq. 共2兲, we obtain the equilibrium free energy as

共10兲

where ␥ is defined by

␥=

冕

⬁

dv␳共v兲.

共11兲

v0

When the amount of oil is large enough to induce the emulsification failure, the free energy does not change any moreeven if the oil is further added. Then the free energy becomes
F/V = − ␭EF␰ − ␥EF − ␳mEF ,共12兲

where the subscript “EF” denotes the corresponding valuesat ␮ = 0. In the following calculations, the integral over the
droplet volume v is replaced with the integral over the radiusr by using the relation dv = 共4␲r2 + 2␲ᐉr兲dr. The integral
range of r spans from 1 to ⬁ since the radius of the smallest
microemulsion droplet is equal to the surfactant size a. We
comment that the present formulation cannot deal with nonglobular structures such as the network or the lamellar
phases since the globular volume v is not well specified inthese structures.
III. RESULTS

Based on the model described in Sec. II, we present theresults of the free energies, phase diagrams, and equilibrium
distributions of microemulsion droplets by changing ¯␬ and c0
systematically. We especially pay attention to the dependence of the emulsification failure boundary and the sphereto-cylinder transition line on the surfactant parameters or thelength of cylindrical droplets. Since the obtained results
qualitatively differ between negative and positive ¯␬, we discuss them separately.
A. Negative saddle-splay modulus „␬¯ < 0…

Some parameters are fixed to ␴ = 0 and ␬ = 1 because
their values do not affect the phase behavior. The others arevaried as c0 = 0.1, 0.6, ¯␬ = −0.6, −0.3, and ᐉ = 0 , 20, 50. In thepresent formulation, the length of the cylindrical droplet ᐉ
cannot be determined from the minimization principle. So
we regard it as a model parameter.
In Fig. 2, we show the plots of the free energy densities
F / V as a function of the oil amount ␾ when the surfactant
amount is fixed to ␰ = 60. In each of the graph, the solid lines
are calculated from Eqs. 共6兲 and 共10兲 by finding the combinations of ␭ and ␮ which satisfy the fixed ␰共=60兲 value. The
dashed lines are obtained by setting ␮ = 0 at which the emulsification failure occurs. The filled circles give the crossing
points between the solid and dashed lines for each ᐉ. Hencethey mark the onset of emulsification failure. Although notshown, each free energy curve is horizontal when ␾ exceeds
the emulsification failure point. At the filled squares, the freeenergy of the spherical droplet crosses that of the cylindrical
droplet. The possible coexistence between the spherical andcylindrical droplet phases17,19,20 is not considered here because we are interested in how the transition lines between
them are dependent on the surfactant parameters or thelength of the cylindrical part. For c0 = 0.1 关Figs. 2共a兲 and2共b兲兴, we see that the spherical droplet phase 共ᐉ = 0; black

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094508-4

J. Chem. Phys. 131, 094508 共2009兲

N. Shimokawa and S. Komura

κ = 0.615

κ = 0.315(a)κ = 0.6
100

(b)

60

40

40

S+O

20

100

φ

0

200

100

φ

0

200

0

(c)
S

60
40

φ

0

40

FIG. 2. Free energies per unit volume F / V as a function of the amount of oil␾ when the amount of surfactant is ␰ = 60. The black, red, and blue solidlines indicate the free energies for ᐉ = 0, 20, and 50, respectively. The dashedlines mark the emulsification failure boundary. The filled circles denote thepoints of emulsification failure. Although not shown, each free energy curve
is horizontal when ␾ exceeds the emulsification failure point. The filledsquares denote the transition points between the spherical and the cylindrical
droplets. The parameters are chosen as 共a兲 c0 = 0.1 and ¯␬ = −0.6, 共b兲 c0 = 0.1
and ¯␬ = −0.3, 共c兲 c0 = 0.6 and ¯␬ = −0.6, and 共d兲 c0 = 0.6 and ¯␬ = −0.3.

lines兲 has the lowest energy compared to the cylindrical
droplet phase 共ᐉ = 20; red line, or ᐉ = 50; blue line兲 at theemulsification failure point. In these cases, the spherical
droplet phase coexists with an excess oil phase. On the other
hand, when ␾ is smaller than the values corresponding to thefilled squares, the cylindrical droplet phase is more stable
than the spherical droplet phase. Although it is not shown in
Figs. 2共a兲 and 2共b兲, the spherical droplet becomes morestable again when the oil amount is made smaller than␾ ⬇ 10 共ᐉ = 20兲.
For c0 = 0.6 as shown in Figs. 2共c兲 and 2共d兲, the spherical
droplet phase coexists with an excess oil phase similar to thecase of c0 = 0.1. However, the sphere-to-cylinder transition
does not occur when c0 = 0.6. Generally speaking, the spherical shape is energetically more favorable than the cylindricalshape when ¯␬ ⬍ 0. Moreover, since the number of cylindrical
droplets for c0 = 0.6 is much larger than that of c0 = 0.1 共seelater Fig. 5兲, the spherical droplet phase is far more stable.
These are the reasons why the cylindrical phase does notshow up here. In Fig. 2共c兲, even a very small amount of oilleads to the emulsification failure of the cylindrical droplet
phase 共hence the red and blue lines are not visible兲. Figure
2共d兲 clearly shows that the free energy of the spherical phaseis always smaller than that of the cylindrical droplet phase.
These sequences of morphological transitions are summarized in Figs. 3 and 4 for ᐉ = 20 and 50, respectively. We
have obtained these phase diagrams by changing the value of␰ between 0 and 100. The boundary line of the emulsification
failure is determined by fixing ␮ = 0 and changing ␭. As theoil amount ␾ is increased from zero in Figs. 3共a兲 and 4共a兲,

S+O

20

0

0

40

20

φ

0

40

20

φ

FIG. 3. Calculated boundaries of emulsification failure and phase transitionlines between the spherical and the cylindrical droplet phases in the case ofᐉ = 20. ␾ is the total amount of oil and ␰ is the total amount of surfactant. S
and C denote the spherical and the cylindrical droplet phases, respectively,
while S + O indicates the coexistence between the spherical droplet phaseand the excess oil. The parameters are chosen as 共a兲 c0 = 0.1 and ¯␬ = −0.6, 共b兲
c0 = 0.1 and ¯␬ = −0.3, 共c兲 c0 = 0.6 and ¯␬ = −0.6, and 共d兲 c0 = 0.6 and ¯␬ = −0.3.

the spherical droplet phase 共S兲 appears first, and it will transform to the cylindrical droplet phase 共C兲 when ␰ is large
enough 共such as ␰ = 60兲. When ␾ is further increased, thecylindrical droplet phase 共C兲 transforms into the sphericalκ = 0.6100

κ = 0.3
100

(a)

80

(b)

80

C
60

C
60

S

40

40

S+O

20
0

S

ξ

20

S

0

100

φ

100

20
0

200

(c)

S+O
0

100

200

φ

100

(d)

80

80

S

60

S

60

ξ

0

(d)

40

S+O

ξ

-20

200

φ

60

ξ

40

c0 = 0.1

φc0 = 0.6

20

100

100

20

0

0

80

ξ

c0 = 0.6

F /V

-10

-10

0

200

φ

80

F /V

0

100

100

(d)

(c)

0

ξ

0

c0 = 0.6

S+O

20

0

0

-20

S

ξ

ξ

c0 = 0.1

F /V

F /V

c0 = 0.1

5

C
S

60

(b)

80

C

10

5

100

(a)

80
10

κ = 0.3

40

40

S+O

0

S+O

20

20

0

20

φ

40

0

0

20

φ

40

FIG. 4. Calculated boundaries of emulsification failure and phase transitionlines between the spherical and the cylindrical droplet phases in the case ofᐉ = 50. ␾ is the total amount of oil and ␰ is the total amount of surfactant. S
and C denote the spherical and the cylindrical droplet phases, respectively,
while S + O indicates the coexistence between the spherical droplet phaseand the excess oil. The parameters are chosen as 共a兲 c0 = 0.1 and ¯␬ = −0.6, 共b兲
c0 = 0.1 and ¯␬ = −0.3, 共c兲 c0 = 0.6 and ¯␬ = −0.6, and 共d兲 c0 = 0.6 and ¯␬ = −0.3.

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094508-5

J. Chem. Phys. 131, 094508 共2009兲

Transitions in globular microemulsions

κ = 0.6

κ = 0.3

0.06

κ = 0.3

0.06(a)κ = 0.6

100

(b)

(a)

100

80

0.02

0.02

60

40

S+O

4

6

r

8

8

10

0

12

6

r

8

10

0

12

2

100

200

φ

0

300

(c)

60

C

100

φ

200

300

(d)

C

60

C+O

40

40

2

0

80

ξ

4

S+O

100

80

c0 = 0.6

4

0

100

(d)

6

ρ

ρ

4

8

(c)

6

c0 = 0.6

2

20

ξ

2

S

ξ

40
20

0

C

60

S

ξ

c0 = 0.1ρρ

c0 = 0.1

C

0.04

0.04

(b)

80

C+O

S

20

20

S+O
2

r

0

3

FIG. 5. Equilibrium size distribution of microemulsion droplets. r is theradius of the microemulsion droplet and ␳ is its number density. The black,
red, and blue lines correspond ᐉ = 0, 20, and 50, respectively. The amount of
surfactant is ␰ = 60. The parameters are chosen as 共a兲 c0 = 0.1 and ¯␬ = −0.6,
共b兲 c0 = 0.1 and ¯␬ = −0.3, 共c兲 c0 = 0.6 and ¯␬ = −0.6, and 共d兲 c0 = 0.6 and¯␬ = −0.3.

droplet phase 共S兲 again. This sequence of sphere共S兲
→ cylinder共C兲 → sphere共S兲 implies a re-entrant transition.
Addition of more oil results in the emulsification failure
共S + O兲. A similar re-entrant transition exists in Figs. 3共b兲 and4共b兲 although the region of the spherical droplet phase andthe sphere-to-cylinder transition line located at ␾ ⬇ 0.1 is not
visible there. However, the re-entrant transition does not occur in Fig. 3共c兲, Fig. 3共d兲 Fig. 4共c兲, or Fig. 4共d兲 because thecylindrical droplet phase does not appear. 关Notice that Figs.
3共c兲 and 3共d兲 are identical to Figs. 4共c兲 and 4共d兲.兴 We pointout that, for all the phase diagrams in Figs. 3 and 4, theexcess oil phase always coexists with the spherical droplet
phase.
We argue next how the changes in the spontaneous curvature c0 and/or the saddle-splay modulus ¯␬ affect the phasebehavior. We see from Figs. 3 and 4 that the region of emulsification failure becomes smaller when either c0 is madesmaller or ¯␬ is made larger. The reason being that the droplet
size becomes larger and the droplet phase is more stable. In
other words, larger droplets can contain more oil even if thetotal amount of surfactant is kept the same. The sphere-tocylinder transition line approaches the emulsification failureboundary as ¯␬ is increased from ⫺0.6 to ⫺0.3. This is because the spherical droplet phase becomes less stable as ¯␬ isincreased. We also find that the region of the cylindricalphase for ᐉ = 50 is larger than that for ᐉ = 20.
Corresponding to the filled circles in Fig. 2, we show in
Fig. 5 the equilibrium size distributions of droplets at theemulsification failure boundary when ␰ = 60. Different colors
indicate different lengths of the cylindrical droplet. The minimum radius of a droplet is r = 1 which is taken to be thesurfactant molecular size a. Hence the droplets with r = 1 do

0

20

40

φ

0

60

0

20

φ

60

40

FIG. 6. Calculated boundaries of emulsification failure and phase transitionlines between the spherical and cylindrical droplet phases in the case of ᐉ
= 20. ␾ is the total amount of oil and ␰ is the total amount of surfactant.
C + O indicates the coexistence between the cylindrical droplet phase and theexcess oil phase, while the other notations are the same with Fig. 3. The
parameters are chosen as 共a兲 c0 = 0.1 and ¯␬ = 0.3, 共b兲 c0 = 0.1 and ¯␬ = 0.6, 共c兲
c0 = 0.6 and ¯␬ = 0.3, and 共d兲 c0 = 0.6 and ¯␬ = 0.6.

not contain any oil. We see that the droplet size r tends to belarger when either c0 is made smaller or ¯␬ is made larger.
This is consistent with the argument in Sec. I where we
discussed that the optimum droplet size is given by 1 / c̃0= 共1 / c0兲关共2␬ + ¯␬兲 / 2␬兴 for a spherical shape at the emulsificaκ = 0.3

κ = 0.6

100

(a)

100

80

(b)

80

C
60
40

S
40

S+O

20
0

C

60

S

ξ

1

0

100

200

φ

100

S+O

20
0

300

(c)

0

100

φ

200

100

300

(d)

80

80
60

C

60

C
ξ

0

ξ

3

ξ

2

r

c0 = 0.11c0 = 0.6

0

C+O

40

C+O

S

20
0

40
20

S+O
0

20

φ

40

60

0

0

20

φ

40

60

FIG. 7. Calculated boundaries of emulsification failure and phase transitionlines between the spherical and the cylindrical droplet phases in the case ofᐉ = 50. ␾ is the total amount of oil and ␰ is the total amount of surfactant.
The notations of the phases are the same with Fig. 6. The parameters arechosen as 共a兲 c0 = 0.1 and ¯␬ = 0.3, 共b兲 c0 = 0.1 and ¯␬ = 0.6, 共c兲 c0 = 0.6 and¯␬ = 0.3, and 共d兲 c0 = 0.6 and ¯␬ = 0.6.

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094508-6

J. Chem. Phys. 131, 094508 共2009兲

N. Shimokawa and S. Komura

κ = 0.3

κ = 0.6

0.015

0.015

(a)

ρ

0.005

0.005

0

10

5

15

r

0

20

(c)

8

10

5

15

r

20

(d)

8

6

ρ

ρ

6

c0 = 0.6

(b)

0.01

ρ

c0 = 0.1

0.01

shows this complex morphological transitions becomes
smaller 关see Fig. 7共c兲兴. This kind of complex sequence ofvarious phases is also alluded in the previous experiments.6,7
In Fig. 8, we show the equilibrium size distributions of
the droplets at the emulsification failure boundary. Similar tothe case of ¯␬ ⬍ 0, the average size is larger when ¯␬ is madelarger and/or c0 is made smaller. The same argument mentioned before applies here as well.

4

4

2

2

0

1

2

r

3

4

0

1

2

r

3

4

FIG. 8. Equilibrium size distribution of microemulsion droplets. r is theradius of the microemulsion droplet and ␳ is its number density. The black,
red, and blue lines correspond ᐉ = 0, 20, and 50, respectively. The amount of
surfactant is ␰ = 60. The parameters are chosen as 共a兲 c0 = 0.1 and ¯␬ = 0.3, 共b兲
c0 = 0.1 and ¯␬ = 0.6, 共c兲 c0 = 0.6 and ¯␬ = 0.3, and 共d兲 c0 = 0.6 and ¯␬ = 0.6.

tion failure boundary. We have checked that the relative peak
position of the size distribution does not change appreciably
even if the ␰ value is varied.
B. Positive saddle-splay modulus „␬¯ > 0…

Next we shall present the results when the saddle-splay
modulus ¯␬ is positive. All the parameters are the same as
before except ¯␬ = 0.3 and 0.6. By calculating the free energies
as in Fig. 2, we obtained the phase diagrams in a similar way.
In Figs. 6 and 7, we show the corresponding phase diagramsfor ᐉ = 20 and 50, respectively. By comparing Figs. 6共a兲 and6共b兲, for example, we see that the region of the cylindricalphase 共C兲 expands when ¯␬ is larger. Hence the sphere-tocylinder transition line approaches the emulsification failureboundary as in the case of ¯␬ ⬍ 0. The cylindrical droplet
phase coexists with the excess oil phase 共C + O兲 for c0 = 0.6
关see Figs. 6共c兲, 6共d兲, 7共c兲, and 7共d兲兴. Such a situation occurs
when the sphere-to-cylinder transition line approaches andcrosses the emulsification failure boundary line. The existence of C + O or S + O region is consistent with the experimental observation.6
When the surfactant amount ␰ is small in Fig. 6共c兲, thespherical droplet phase undergoes the emulsification failure
共S + O兲 as ␾ is increased. When ␰ is large, the cylindrical
droplet phase coexists with the excess oil phase 共C + O兲. By
increasing the oil amount ␾ while fixing the surfactant
amount to ␰ ⬇ 20, we see a rather complicated sequence ofmorphological transitions. First the spherical droplet phase
共S兲 transforms into the cylindrical droplet phase 共C兲 which
undergoes the emulsification failure 共C + O兲. As ␾ is further
increased, the system exhibits the spherical droplet phase 共S兲
which again undergoes the emulsification failure 共S + O兲. For
longer cylindrical droplets 共ᐉ = 50兲, the region of ␰ which

IV. DISCUSSION

Several points merit further discussion. In Figs. 3共a兲,
3共b兲, 4共a兲, and 4共b兲, we have explained that the system exhibits the re-entrant transition 共sphere→ cylinder→ sphere兲
as the oil amount ␾ is increased. The mechanism of thistransition can be understood as follows. In the above phase
diagrams, we have shown that the spherical droplet phase ismore stable than the cylindrical droplet phase when ␾ iseither small or large. This is because a spherical structure ismore favored when ¯␬ is negative. In the intermediate oilconcentration, the cylindrical droplet phase appears for thefollowing reason. We have assumed that the surfactant molecules form either monolayers or spherical micelles. In general, the formation of spherical micelles having a large curvature costs energy especially when c0 is small. In theintermediate oil concentration, the number of created micelles associated with spherical oil droplets is larger than thatassociated with cylindrical oil droplets. In fact, spherical
droplets need less surfactant to incorporate a fixed amount ofoil so that the remaining surfactant molecules inevitably
form micelles which cost high energy. As a whole, the cylindrical droplet phase can lower its energy with respect to thespherical droplet phase. It should be noted that, close to theemulsification failure boundary, almost all of the surfactant
molecules are used to cover oil droplets so that micelles
hardly exist.
Next we discuss why the cylindrical droplet phase undergoes a direct transition to the emulsification failure when¯␬ ⬎ 0 关see Figs. 6共c兲, 6共d兲, 7共c兲, and 7共d兲兴. In order to understand it, we look at the size distributions of droplets given by
Fig. 8共d兲. The peak positions are located at r ⬇ 2.2 for ᐉ = 0
共spheres兲 or r ⬇ 1 for ᐉ = 20 and 50 共cylinders兲. We can also
check that the optimum radius obtained by the minimizationof the curvature energy coincides with the corresponding
peak position of the size distribution. Since most of the cylindrical droplets has r ⬇ 1, they cannot incorporate a large
amount of oil. Hence the number of cylindrical droplets becomes large as seen in Fig. 8共d兲. Due to the same mechanism
described in the previous paragraph, the cylindrical droplet
phase becomes relatively stable compared with the spherical
droplet phase. Moreover, cylindrical droplets are energetically favorable when ¯␬ is positive. According to these reasons, the cylindrical droplet phase can undergo a direct transition to the emulsification failure without changing into thespherical droplet phase.
We discuss here how our results are related to the existing experimental observations. Although many phase diagrams are explored in experimental works, it is not straightforward to compare them directly with ours because their

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094508-7

J. Chem. Phys. 131, 094508 共2009兲

Transitions in globular microemulsions

axes are often different. Nevertheless, some qualitative comparisons are possible. In general, the spontaneous curvature
c0 decreases as the temperature is increased. In some experiments, it was shown that the emulsification failure region
becomes smaller and the droplet size becomes larger, as thetemperature is increased.6,7,13,14 The phase diagrams and theequilibrium size distributions obtained from our model exhibit the same tendencies when c0 is decreased. On the other
hand, the values of the bending modulus and the saddlesplay modulus can be deduced from interfacial tension measurements or neutron spin-echo spectroscopy.5,26 Most of theexperimental results suggest that ¯␬ takes a negative value sothat the cylindrical droplet phase becomes more stable asc0 is decreased. For a ternary mixture ofdodecane/ C12E5 / water,27 it was shown that the transition
from the spherical droplet phase to the cylindrical droplet
phase takes place as the temperature is increased. When ¯␬⬍ 0 共see Figs. 3 and 4兲, the cylindrical droplet phase does notexist in the phase diagrams for c0 = 0.6. However, the cylindrical droplet phase becomes more stable than the spherical
droplet phase as c0 is decreased. Hence our result predicts
that the transition from the sphere to cylinder may occur as
the temperature is increased. This is consistent with the experimental observation.27
In the present theory, we treated the length of the cylindrical part ᐉ as a parameter. Blokhuis and Sager21 assumed
that the polydispersity in the cylinder length obeys the
Schultz distribution. This assumption is well supported by
the experiment.28 However, it was shown that the phase diagrams hardly change even if the average cylinder length isused instead of taking into account the polydispersity
distribution.21 Moreover, one of the main purposes of thispaper is to reveal the dependence of the transition lines on
the cylinder length. These are the reasons why we did notminimize the free energy with respect to ᐉ.
Finally, we compare our results with the previous theories. Safran et al.18 analyzed the relative stability among
spherical, cylindrical, and planar monolayers and pointed outthat the ratio ¯␬ / ␬ is an important parameter. While restricting only to the case of ¯␬ ⬍ 0, they showed that the sphereto-cylinder transition approaches to the emulsification failureboundary when ¯␬ is increased. This is in agreement with ourphase diagrams given such as in Figs. 3共a兲 and 3共b兲. They
also showed that only the spherical droplet phase coexists
with the excess oil phase, which is again in agreement with
our result. When ¯␬ ⬎ 0, a direct transition from the cylindrical droplet phase to the emulsification failure was anticipatedin Refs. 20 and 21. The corresponding phase diagram is, forexample, Fig. 6共d兲 in our paper. We also mention that Safran
et al. argued the coexistence between the spherical and cylindrical droplet phases,17,19,20 and the volume fractions ofsphere and cylinder in this coexistence region was calculated
later in Ref. 21. However, since we are interested in thedependence of the transition lines on the surfactant parameters or the cylinder length, we did not take into account thepossible phase coexistence for clarity.

V. CONCLUSION

In this paper, we have considered the emulsification failure of the microemulsion droplet phases. By taking into account both the cylindrical and the spherical droplet phases,
we have obtained both the sphere-to-cylinder transition line
and the emulsification failure boundary. We have discussed
the effects of the saddle-splay modulus ¯␬ and the spontaneous curvature c0 on the phase behavior. When ¯␬ ⬍ 0, thespherical droplet phase coexists with the excess oil phase. A
re-entrant transition 共sphere→ cylinder→ sphere兲 is expected
for small c0 cases. When ¯␬ ⬎ 0, a direct transition from thecylindrical droplet phase to the emulsification failure would
occur as long as c0 is large. In some case, emulsification
failure may occur twice as the oil amount is increased. In
general, the sphere-to-cylinder transition line approaches theemulsification failure boundary as ¯␬ is increased.

ACKNOWLEDGMENTS

N.S. is supported by a Japan Society for the Promotionof Science Research Fellowship for Young Scientists 共Grant
No. 20–10527兲 from the Ministry of Education,
Culture, Sports, Science and Technology of Japan. S.K.
thanks the support by KAKENHI 共Grant-in-Aid for
Scientific Research兲 on Priority Areas “Soft Matter Physics”
and Grant No. 21540420 from the Ministry of Education,
Culture, Sports, Science and Technology of Japan.
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