JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 6
8 FEBRUARY 1999
Periodic surfaces and cubic phases in mixtures of oil, water,
and surfactant
Alina Ciach and Robert Hołysta)
Institute of Physical Chemistry PAS and College of Science, Dept. III, Kasprzaka 44/52,
01224 Warsaw, Poland
~Received 9 December 1997; accepted 4 November 1998!
We study a ternary mixture of oil, water, and surfactant in the case of equal volume fractions of oil
and water using the Landau–Ginzburg model derived from a lattice model of this ternary mixture.
We concentrate on a phase region close to a coexistence line between microemulsion and cubic
phases. In our model the bicontinuous cubic phases exist in a narrow window of the volume fraction
of surfactant r s '0.6. The sequence of phase transitions is L→G→D→ P→C as we increase r s
along the cubic-microemulsion bifurcation line. Here L stands for the lamellar phase and C for the
cubic micellar phase. The gyroid G, primitive P, and diamond D phases are all bicontinuous. The
transitions weakly depend on the temperature. The increase of r s is accompanied by the increase of
the surface area per unit volume. In the case of fluctuating monolayers the interface is diffused and
the average area of the monolayer per unit volume is larger than the ‘‘projected area,’’ i.e., the area
of the surface describing the average position of the monolayer, per unit volume. The effect is the
strongest in the L and the weakest in the C structure. © 1999 American Institute of Physics.
@S0021-9606~99!51306-8#
I. INTRODUCTION
bic cell can yield partial information about a topology of the
structure. One can expect a small unit cell and the small
surface area per side of the unit cell in structures of the
simple topology8 and the large values for the structures of
complex topology. Typically the surface area per side of the
cubic cell is between 2 and 4 for simple topology surfaces
such as P,D,G,I-W P 6 , O-CTO 6 etc., but it can be larger
than 7 for the complex topology surfaces.9–11
In binary mixtures of the water, the surfactants or the
lipids the most common structure is the gyroid one, G, existing usually on the phase diagram between the hexagonal
and lamellar mesophases. This structure has been observed
in a very large number of systems1,12–15 and in the computer
simulations of the systems.16 The G phase is found at rather
high surfactant concentrations, usually much above 50% by
weight. Other cubic phases even at very low concentration of
surfactants can be found in ternary mixtures ~with water and
oil!.8,17,18 For the didodecyldimethylammonium bromide,
water, and styrene system, the periodic surfaces are found
over the huge range of water fraction from 11% to over 80%.
Most of the studies concentrated on the cases with large
amounts of water and surfactant and the influence of added
oil on the phase transitions. It was shown that the emerging
phases depend on the properties of the oil, i.e., whether it
penetrates the surfactant or swells the bilayer. In the highly
asymmetric case of a large amount of the water and the surfactant and a small amount of the oil it has been found that
with increasing the volume fraction of water one finds the
following progression of cubic phases: G→D→ P. These
studies are time consuming due to the very long equilibration
times of weeks or even months. Nevertheless the experimental data are very rich in this case.19–23
Here we concentrate our theoretical studies concerning
A periodic surface is a surface that moves onto itself
under a unit translation in one, two, or three coordinate directions similarly as in a periodic arrangement of atoms in
regular crystals. Triply periodic minimal surfaces are periodic in all three directions and are in addition characterized
by the zero mean curvature at each point of the surface. The
first triply periodic minimal surface ~primitive, P, surface!
was discovered by Herman Schwarz in 1865. The interest in
periodic surfaces in this century was due to the experimental
observation ~Luzzati et al.1! that bilayers of lipids ~or surfactants! in water solutions form at suitable thermodynamic
conditions ordered bicontinuous structures. To describe these
mesophases the ideas of minimal surfaces and related hyperbolic structures were next used by Larsson, and Hyde
et al.2–5 Also, new triply periodic minimal surfaces ~among
them the gyroid, G, surface! were discovered by a mathematician Alan Schoen.6 If we draw a surface through the middle
of the triply periodic lipid ~or surfactant! bilayer, it divides
the volume into two disjoint subvolumes each continuous in
the whole volume of the system.7 Therefore, the name bicontinuous has been given for such structures.
The x-ray scattering provides information about the
symmetry of the structure and, therefore, discerns between
different periodic surfaces. Here we concentrate on the cubic
symmetries. The set of the reflections for the P cubic structure (Im3m symmetry! index to &: A4: A6: A10: A12:
A14: A16, that of the D structure ( Pn3m symmetry! index to
&:): A4: A6: A8: A9: A10, and that of the G structure
(Ia3d symmetry! index to A6: A8: A14: A16: A20: A22: A24.
Finally, the measurements of the surface area inside the cua!
Electronic mail: [email protected]
0021-9606/99/110(6)/3207/8/$15.00
3207
© 1999 American Institute of Physics
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3208
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
ordered cubic phases on the largely unexplored region of
phase diagram of the ternary mixture of water, oil, and surfactant, where the volume fractions of oil and water are comparable. It is well known that bicontinuous microemulsions
are formed in such systems. In bicontinuous microemulsions
the surface covered with the surfactant divides volume into
the water rich and the oil rich subvolumes. The diffusion
measurements provide direct information about the bicontinuity of the structure. The pulsed gradient nuclear magnetic
resonance ~NMR! self-diffusion technique gives the selfdiffusion rates of all the components in the structure,24 providing a direct check on the continuity and the extension of a
region occupied by the components. If any of the components of the system is closed in a finite ~small! volume, then
its effective diffusion constant ~measured as a mean-squared
displacement divided by time! goes to zero, whereas in a
continuous structure the effective diffusion coefficient is
nonzero.
The range of order in microemulsions is comparable to
the typical length of the structure ~domain size!. Topological
properties of the surfactant monolayers resemble those of
periodic surfaces. It is still not clear, however, whether the
transition between the bicontinuous microemulsion and the
ordered bicontinuous cubic phases occurs in nature. When
the volume fractions of the oil and the water are equal, one
finds the cubic phases in a narrow window of the surfactant
concentration around 0.5 weight fraction.19,20 However, it is
not known whether these phases are bicontinuous. No experimental evidence has shown that there exist bicontinuous
cubic phases with the ordered surfactant monolayer, rather
than bilayer, forming the periodic surface. We study the possibility of such ordering transition between microemulsion
~random isotropic monolayer! and ordered bicontinuous cubic phases ~ordered periodic monolayer! within Landau–
Ginzburg approach in this work.
For the comparable volume fractions of oil and water,
bicontinuous microemulsions and the other phases with vanishing spontaneous curvature are stable. The lack of an oil–
water symmetry is for this part of phase diagram not of primary importance, since the surfactant monolayer is not
biased towards either oil or water occupied region. Effectively oil and water play symmetrical roles with respect to
the surfactant in this limited part of the phase diagram.
Hence for an equal oil and water volume fractions one can
describe the system by an oil–water symmetric model.
The Landau–Ginzburg model used here25 has been derived from the lattice model of the ternary mixture.26 A volume occupied by a single molecule is fixed in the lattice
model. Due to the oil–water symmetry in the model the
spontaneous curvature is zero. In the model the important
field is the local oil–water volume fraction difference and the
fields describing local assembling and orientational ordering
of amphiphiles. The model differs strongly from the theoretical description introduced in terms of the geometrical properties of the surfactant molecules and the surface covered by
them, where the various phase transitions are interpreted in
terms of the geometry of the system and change of the curvature or the surface area per head of surfactants.27–29 Nevertheless we show a connection between the results of the
A. Ciach and R. Hołyst
Landau–Ginzburg model and the geometry of the surface
dividing the volume into oil-rich and water-rich regions. In
our model we compute the sequence of phase transitions between G, D, and P phases as a function of r s .
The paper is organized as follows. In Sec. II we present
the equations describing different phases and discuss their
geometrical properties. In Sec. III we discuss the model and
present its simplified form near the coexistence line between
the microemulsion and cubic phases. In Sec. IV we show the
results and discuss the calculated sequence of phase transitions. The conclusions are also contained in Sec. IV.
II. THE L, P, D, G, AND C STRUCTURES STUDIED IN
THE MODEL
Let f~r! denote the scalar field representing the difference between oil and water volume fraction at the point r.
The structures studied in this paper can be represented by the
following Fourier series:
f ~ r! 5 ( A ~ k ! cos@ 2 p k•r2 a ~ k !# ,
k
~2.1!
where k describes the reciprocal lattice vectors for a given
lattice, a (k) is a phase shift, and A(k) is an amplitude associated with a modulus, k, of a given k-vector. Close to the
bifurcation line ~a line of continuous phase transition and/or
divergence of the static structure factor! the amplitudes are
very small and it is justified to use only the first term in the
Fourier series. The lamellar structure, L, is given by
f ~ r! 5A L cos~ Z ! .
~2.2!
The P structure is described by
f ~ r! 5A P ~ cos~ X ! 1cos~ Y ! 1cos~ Z !! ,
~2.3!
the D structure is given by
f ~ r! 5A D ~ cos~ X ! cos~ Y ! cos~ Z ! 2sin~ X ! sin~ Y ! sin~ Z !! ,
~2.4!
while the gyroid G structure by
f ~ r! 5A G ~ sin~ X ! cos~ Y ! 1sin~ Y ! cos~ Z ! 1cos~ X ! sin~ Z !! .
~2.5!
Finally the cubic micellar structure, C, is represented by
f ~ r! 5A C cos~ X ! cos~ Y ! cos~ Z ! .
~2.6!
Here X52 p x/d, Y 52 p y/d, Z52 p z/d, A i , ~i is L, P, D,
G, or C! is an amplitude and d is the size of the unit cell. Due
to the symmetry between oil and water the dividing surface
where the surfactant is located is given by
f ~ r! 50.
~2.7!
The surfaces defined by Eqs. ~2.3!–~2.5! are called the nodal
surfaces.30–34 Although the nodal surfaces are neither minimal nor constant mean curvature they can be used as an
ansatz for such surfaces.35 For example, the topology of the
minimal surface and its symmetry are exactly the same as the
ones of the corresponding nodal surfaces. The geometrical
properties such as the surface area per side of the unit cell
(Sd) ~here S is the surface area per unit volume and d is the
size of the unit cell! are very well represented by the corre-
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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
A. Ciach and R. Hołyst
3209
FIG. 1. The nodal P primitive surface given by Eqs. ~3.3! and ~3.7! inside
the unit cell of the primitive bicontinuous cubic structure ~a view along a
diagonal!.
FIG. 3. The nodal G gyroid surface given by Eqs. ~3.5! and ~3.7! inside the
unit cell of the gyroid bicontinuous cubic structure ~a view along a diagonal!.
sponding properties of these nodal surfaces. The quantity
mentioned above has the value very close to the one for the
minimal surface ~differences about 0.5%!. We find using Eq.
~2.7! together with Eqs. ~2.3!–~2.5! the following values for
these nodal surfaces: for P, Sd52.353, for the G surface
Sd53.092 and for the D surface we have Sd53.839. In
Figs. 1–4 we show the nodal P, D, and G surfaces and our
cubic micellar C surface in the unit cell of the given structure. In the actual calculations we will use the Fourier amplitudes of the structures.25
ever, cubic phases, present in mixtures with strong surfactants, are not stable in this model.9–11 To study such phases
one has to consider more complicated LG models, in which
additional order–parameter, describing orientational ordering
of surfactant particles, is present. In case of binary mixtures
such model was introduced in Ref. 37 whereas for ternary
mixtures a model in which cubic phases are stable was introduced and studied in Ref. 25. Important advantage of the
latter model is the fact that all the coupling constants are
expressed in terms of the surfactant volume fraction r s , temperature, and a single phenomenological parameter g describing the amphiphilicity of the surfactant through the
strength of interparticle interactions. Small g corresponds to
weak, and large g to strong surfactants. No parameters in this
model are fitted, and all the calculated quantities are expressed in terms of directly measured quantities and a single
phenomenological parameter specifying the material properties of the system. No additional assumptions concerning for-
III. THE LANDAU–GINZBURG „LG… MODEL NEAR
THE BIFURCATION LINE
Many properties of mixtures with weak surfactants are
well described by the elegant Gompper–Schick ~GS!
model,36 in which the free energy is a functional of the local
difference between volume fractions of oil and water. How-
FIG. 2. The nodal D diamond surface given by Eqs. ~3.4! and ~3.7! inside
the unit cell of the double diamond bicontinuous cubic structure ~a view
along a diagonal!.
FIG. 4. The nodal C surface given by Eqs. ~3.6! and ~3.7! inside the unit cell
of the micellar cubic structure ~a view along a diagonal!.
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3210
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
A. Ciach and R. Hołyst
mation of the monolayer or its properties are made. Within
this LG model one can calculate the positions of surfactant
surfaces and their properties for different r s , and draw conclusions about such physical properties, as the sequence of
cubic phases or the effect of diffused ~or delocalized! oil–
water interfaces on the average surface area. Please note that
S is a projected area per unit volume and in general is not
equal to the average area per unit volume S/V of the
surfactant-occupied surface. The projected area is the area of
the surface that describes the average position of the interface between water rich and oil rich domains @given by Eq.
~2.7!#.
Instead of introducing the LG model on symmetry
grounds, in Ref. 25~a! the free—energy functional is derived
from the lattice microscopic model26 in which only two parameters characterize the interactions in the case of the oil–
water symmetry: b is the strength of the water–water ~oil–
oil! interaction, and c describes interaction between the water
~oil! and the amphiphiles ~surfactants!. Eventually only a
combination of b and c enters as a single interaction parameter g 5(2c/b) 2 . Interaction between the amphiphiles and
ordinary particles is proportional to a scalar product between
the orientation of the amphiphile and the distance between
the particles. In this way the amphiphilicity is explicitly
taken into account. The interactions between amphiphiles in
the simplest version of the model are neglected.
In the model of Ref. 26 the microscopic density distributions in the case of close packing of the molecules are
given by r̂ a (r) where a denotes oil, water, or surfactant in
different orientations and r denotes a lattice site. r̂ a (r)
51(0) if the site r is ~is not! occupied by the component a.
This microscopic model can be investigated in the meanfield ~MF! approximation, in which microscopic configurations r̂ a (r) occur with a probability ;exp@2bHMF# , where
for any microscopic state the MF—Hamiltonian H MF is
given by
H
@ r̂ a ~ r!# 5
MF
(r (a c a~ r!~ r̂ a~ r! 2
1
2
r a ~ r!! .
~3.1!
r a (r) is the average density ~introduced here to compensate for double counting of pairs of molecules! and c a (r) is
the mean field felt by the particle of the kind a at r, resulting
from interactions with the remaining molecules. In other
words, H MF@ r̂ a (r) # is equal to the total energy of the microscopic state @ r̂ a (r) # in a hypothetic system in which every
particle experiences only an external field c a (r), which is a
functional of the average density.26 The strength of this hypothetic external field at a given point is equal to the field
provided by the rest of the system, as if it were in the hypothetic state corresponding to the minimum of the grand thermodynamical potential.26~b! In the Weiss-type formulation of
the mean field, the average density is given by
r a 0 ~ r0 ! 5const
(
@ r̂ a ~ r!#
r̂ a 0 ~ r0 ! exp~ 2 b H
A particular microscopic state in such ordered phases appears with a probability ;exp(2bHMF@ r̂ a (r) # ) which
strongly depends on the average densities @through c a (r)#.
For the structures with sharply localized average states the
microscopic configurations different than the average solution appear with negligible probability. However, if r a (r)
are slowly varying functions, as is the case close to secondorder transitions,26 then many microscopic configurations occur with comparable probability @see Eq. ~3.2!#.
In an equivalent formulation of the MF, in order to find
the equilibrium state, i.e., the average density, with the probability distribution ;exp@2bHMF# , one minimizes the MF
approximation for the grand thermodynamical potential.26
Close to the continuous transition one can further simplify
the analysis, by using the continuous approximation for the
lattice model discussed above. The LG functional can be
defined as a continuous approximation for the MF thermodynamical potential. In Ref. 25~a! the LG functional corresponding to the microscopic model described above is derived in a standard way and is found to be a functional of
three-order parameter ~OP! fields: f~r!, the concentration
difference between oil and water, r~r!, the deviation of the
average of the local density of surfactant from the global
value, and a vector field u~r! describing the orientational
ordering of the amphiphiles.
The resulting functional assumes the form
V eff5 ~ V 2 1V int! b,
with
V 25
dr@ 21 a 2 f 2 1 21 ~ ¹ f ! 2 1 21 a 2 r 2 1 12 ~ ¹ r ! 2
~3.3b!
and
V int5
E
dr@ 3!1 ~a3r31b3f2r1c3uuu 2 r !
1 4!1 ~a4f41a4r41b4f2r21c4r2uuu 2 1A 4 u uu 4 !# .
~3.3c!
All the coupling constants can be expressed in terms of the
average surfactant volume fraction r s , temperature t
5kT/b, and a parameter describing the amphiphilicity g
5(2c/b) 2 . The grand-thermodynamical potential and the
temperature are calculated in units of b. Macroscopically b is
related to the critical temperature of oil–water separation by
kT c 5d(12 r s )b. The explicit expressions for the coupling
constants which are used here for bifurcation analysis are
a 2 52
S
F
a 2 52
@ r̂ a ~ r!# ! ,
which is to be solved self-consistently.
For ordered states the average densities r a (r) are not
constant, but rather have some kind of oscillatory behavior.
E
1 12 @ u uu 2 1 ~ ¹•u! 2 1 ~ ¹3u! 2 # 2Ju•¹ f #
MF
~3.2!
~3.3a!
J5
D
t
2d ,
12 r s
G
t
2d ,
r s ~ 12 r s !
S D
b 35
~3.4a!
2 r sg
3t
~3.4b!
1/2
,
~3.4c!
6& t
,
~ 12 r s ! 2
~3.4d!
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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
c 3 52
a 45
3&
,
rs
8t
~ 12 r s ! 3
A 45
A. Ciach and R. Hołyst
3211
~3.4e!
~3.4f!
,
18
.
5trs
~3.4g!
The other coupling constants which are not needed in the
present paper are given in Ref. 25~a!.
The length unit in Eq. ~3.3! is by construction of the LG
model equal to the lattice constant a[1 of the original
model. In the lattice model it was assumed that a lattice cell
is occupied by a single particle ~in the case of water by a
cluster of particles!. Hence the lattice constant is identified
with the linear size of the amphiphile, and thus the length
unit used here is ;25 Å. In our mesoscopic description the
volume is measured in units of the volume occupied by a
surfactant molecule, and the area is measured in units of the
area occupied by an amphiphile. In other words, in our
model the area of the monolayer is the dimensionless quantity equal to the number of amphiphiles residing on the
monolayer. Hence, it should be identified with the area rescaled by the surfactant parameter of the corresponding structure ~see the discussion in Ref. 38!, when comparing our
results with those of the microscopic, geometrical models.38
The rescaled area of the monolayer per unit volume S is
equal to the surfactant volume fraction if no surfactant molecules occur outside the monolayer.38
An example of a phase diagram, calculated in Ref. 25~a!,
is shown in Fig. 5~A! for g 550 ~strong surfactant!. For low
surfactant-volume fraction there is a continuous transition
between the homogeneous fluid and coexisting homogeneous
oil- and water-rich phases, given by a 2 50. Next there is a
Lifshitz point, at r s 5 r Ls , with r Ls '0.1 for g 550, at which
the Lifshitz line meets the bifurcation line. The Lifshitz line
separates the uniform phase into a structureless region
~above! and microemulsion ~below!. Here by microemulsion
we mean macroscopically uniform phase in which water–
water structure factor assumes maximum for the wavenumber kÞ0. For r s . r Ls the instability of the uniform phase
for the wave-number k5k b Þ0 occurs at temperatures higher
than the instability with respect to oil–water separation (a 2
50 and k50) and is given by25
e 2 [ ~ 11a 2 2J 2 12 Aa 2 ! /450, and
k b 5a 1/4
2 .
~3.5!
The bifurcation line with k5k b Þ0 defines continuous transition between microemulsion and different ordered phases.
In Ref. 25~a! it was found that the continuous transition to
the ordered phases terminates at the tricritical point indicated
by the last cross on the right in Fig. 5~A!. For r s larger than
at the tricritical point the transition between microemulsion
and ordered phases becomes first order within the Landau
model. The different ordered phases coexist below the bifurcation line. The first-order transition lines meet the continuous transition to the microemulsion at points indicated by the
crosses in Fig. 5. Fig. 5~B! shows a portion of the bifurcation
line corresponding to the stability region of the cubic phases.
FIG. 5. ~A! The bifurcation ~solid! line between a homogeneous fluid and
cubic phases. t is the dimensionless temperature @see below Eq. ~3.3!# and
r s is the surfactant volume fraction ~the volume fraction of oil and water are
equal!. The dashed line is the Lifshitz line below which the water–water
structure factor assumes maximum for a wave-number kÞ0, as in microemulsions. o/w denotes the coexisting oil and water-rich phases. lc denotes
liquid–crystalline phases, separated by crosses. The last cross is the tricritical point separating the continuous and the first-order phase transitions. ~B!
The part of the bifurcation line @shown in full in ~A!# between the microemulsion and the cubic phases. t is the dimensionless temperature @see
below Eq. ~3.3!# and r s is the surfactant-volume fraction. L, G, D, P, and C
denote the lamellar, gyroid, diamond, primitive, and cubic micellar structures, respectively. Along the bifurcation line the static structure factor diverges for the wave vector k b .
The locations of the coexistence regions for lower temperatures are not indicated. We limit ourselves to the vicinity of
the continuous transition, where analytical results can be obtained. Much below the continuous transition extensive numerical calculations are necessary.
The full MF analysis of phase diagrams for weak as well
as for strong surfactants has been performed in Ref. 26~c! for
the simplified version of the present model. The interactions
in the model of Ref. 26~c! are the same as in the present case,
but the locations of the particles are restricted to sites of a
simple-cubic lattice and, unlike in the present model, the
orientations of amphiphiles are restricted to six directions
compatible with the lattice structure. Because of this restriction the ordered phases other than the D phase are not stable,
as they would correspond to ‘‘forbidden’’ orientations of
amphiphiles ~for example, in the case of the lamellar phase
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3212
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
A. Ciach and R. Hołyst
the direction of oscillations would correspond to the diagonal
of the lattice cell!.26~c! For the uniform phases both models
give the same results, and the general features of the phase
diagrams of the full and the simplified model should be similar. For strong surfactants @Fig. 5~b! of Ref. 26~c!# it was
found that for low temperatures there are large three- and
two-phase regions; for example,26~c! for temperatures t ,1,
the oil-, water-, and D-phase coexistence occurs for 0, r s
,0.7. For higher temperatures ( t .1.5), however, a substantial part of the phase space is occupied by the ordered D
phase, as determined by the bifurcation analysis in perfect
agreement with the full MF study.26~c! The results obtained in
the simplified model indicate that also in the full model substantial regions of stabilities of the ordered phases are to be
expected below the bifurcation line, and that only for significantly lower temperatures the large two- and three-phase coexistence regions occur. This feature of the phase diagrams
indicates that the bifurcation analysis is not only limited to
the immediate vicinity of the bifurcation line, but should
apply to a rather extended temperature region.
We should also stress that in this model the pure surfactant system does not undergo ordering phase transition, because in the model introduced in Ref. 25~a!, the interactions
between amphiphiles are neglected. For a very high surfactant concentration the results obtained in this model are not
expected to agree with experiments. The amphiphiles do not
order by themselves, thus if there is too much surfactant, the
ordering effect of interactions between ordinary particles and
amphiphiles becomes too weak and the system becomes disordered, whereas in reality the interactions between the amphiphiles support formation of a lamellar phase for very high
surfactant volume fraction.
We extend here the bifurcation analysis of Ref. 25~a!,
and consider in addition to the already studied structures, the
gyroid phase. From now on we concentrate on r s . r Ls , for
which the instability of the disordered phase corresponds to
k5k b .0. Close and below the bifurcation line t 5 t b ( r s ),
given explicitly by Eq. ~3.5!, V has a much simpler form, if
the contributions of order O( e 6 ), are neglected.25 In equilibrium V assumes minimum. This condition allows to express
r and u in terms of f (k)5O( e ) corresponding to bifurcation, and
V5
2k 2b
(k 11k b
2
2
2 e u f̃ ~ k ! u 1
1
8
d Kr
4! k1 ,k2 ,k3 ,k4
(
S( D
4
i
ki
4
3@ A ~ $ k̂i % ! 2G ~ $ ki % !! ]
)i f̃ ~ ki ! 1O ~ e 6 ! .
~3.6!
In the above k b 5a 1/4
2 is the wave number corresponding to
bifurcation,25~a! and
A ~ $ k̂i % ! 5a 4 1A 4 a 2 ~ k̂1 •k̂2 !~ k̂3 •k̂4 ! ,
G ~ $ ki % ! 5
~ b 3 2c 3 k1 •k2 !~ b 3 2c 3 k3 2k4 !
.
3 ~ a 2 1 u k1 1k2 u 2 !
~3.7a!
~3.7b!
In Eq. ~3.6! ( 8 is a summation over vectors such that u ki u
5k b .
The stable phase is the one giving the lowest value of
Eq. ~3.6! for the given value of r s and for u ku 5k b given by
Eq. ~3.5!.
The Fourier amplitudes of the structures defined in Eqs.
~2.2!–~2.6! just below the bifurcation can be written in a
form
F
m
m
j
j
G
f̃ ~ k! 5F w ( d Kr ~ k2k b p̂ j ! 1w * ( d Kr ~ k1k b p̂ j ! .
~3.8!
Here w * denotes the complex conjugate to w. For the structures L, P, and C w51, for D w511i and for G w5i. The
number and orientations of the unit vectors p̂ j are different
for different structures. For the L phase there is m51 vector
in the direction of oscillations. For the D and C phases m
54 and p̂ j form tetrahedron. For the G phase m56 and each
unit vector p̂ j is parallel to one of the 6 diagonals of the three
adjacent sides of the cube. The size of the unit cell is d
52 p /k b for the L and P phases; d52 p )/k b for the D and
C phases and d52 p &/k b for the G phase. In Ref. 25~a! the
hexagonal phase was considered in addition to the cubic
phases listed in the Sec. II. It was shown that this phase is
not stable in this part of the phase diagram, that is close to
the stability region of microemulsion. Hence, we do not consider this phase in the present work.
Just below the bifurcation f~r! is a slowly varying function of the position. For example, in the simplest, lamellar
phase with the modulations in a ẑ direction and for the surfactant surface located at z50, f (z); e sin(2pz/kb). The
density of surfactant is given by ; e cos(2pz/kb), hence it
deviates from zero in an extended region. The width of the
surfactant-occupied interface is very large.
IV. RESULTS, THEIR GEOMETRICAL
INTERPRETATION AND DISCUSSION
We have calculated V given by Eq. ~3.6! for the structures described in the Sec. II and the portion of the phase
diagram corresponding to stable cubic phases is shown in
Fig. 5~b! for strong surfactants ( g 550). For weaker surfactants ~but still strong, i.e., with water–amphiphile interactions stronger than the water–water interactions! the sequence of phases is the same, L→G→D→ P→C. For
sufficiently strong surfactants, for which cubic phases are
stable, the only dependence of the phase diagram on the
strength of surfactant is the extent of the region of stability of
the lamellar phase. It grows with the strength of the surfactant. For surfactants weaker than in the case shown in Fig. 5
the cubic phases occur for somewhat lower surfactantvolume fraction, for example, for g 516 cubic phases appear
for r s '0.45. It is worth noting that the surfactant-volume
fraction in cubic phases found in this model agrees very well
with the surfactant-volume fraction for which unidentified
cubic phases are stable in ternary mixtures with equal oil and
water volume fractions for C i E j with (i, j)5(8,4) or
~10,5!.19 It is also important to note that the bicontinuous
cubic phases exist in a very narrow range of surfactantvolume fractions in our model, and that they transform into
the cubic micellar phase when r s is increased.
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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
FIG. 6. The projected rescaled surface area @i.e., the area multiplied by the
surfactant parameter ~Ref. 38!# per unit volume S divided by the surfactant
volume fraction for different structures along the bifurcation line as a function of surfactant volume fraction r s . Please note that due to the geometrical constraints this quantity cannot exceed the length of the surfactant l .
Here we set l 51 for convenience.
In a structure with all surfactant molecules located at
monolayers, the volume fraction of surfactant should be
equal to the average surface area times the surfactant parameter, times the width of the monolayer divided by the volume, i.e., r s 5S l /V, where S is the surface area rescaled by
the surfactant parameter. This is true for a single surface with
no intersections. In real systems the microscopic configurations of the monolayers @which are taken into account in the
derivation of the Landau–Ginzburg model in Eq. ~3.2!# can
differ from the average configuration and the rescaled average surface area per unit volume S/V is larger than the rescaled ‘‘projected’’ surface area per unit volume S. The surfactant density averaged over these configurations is not a
sharply localized function, as for a single configuration of a
monolayer. It is a smooth function, nonvanishing in a region
of width comparable to the standard deviation from the average position of the monolayer. The projected surface corresponds to the average location of the interface between
oil-rich and water-rich domains and in our case is given by
Eq. ~2.7! @ f (r)50 # and Eqs. ~2.2!–~2.6!. Hence we expect
S/ r s < l 21 .
The ratio S/ r s , calculated for different phases below the
bifurcation is shown in Fig. 6. Recall that S is the rescaled
surface area per unit volume, i.e., the area multiplied by the
surfactant parameter.38 The rescaled surface area per volume
is an increasing function of the surfactant-volume fraction
and it determines the sequence of phases. Moreover, we have
found that the deviations of the value S/ r s from unity ~for
convenience we set l 51 in our model! are different in different phases. In the model considered here within the MF
approximation the surfactant-occupied interface is very diffuse close to the second-order transition. As discussed in
Sec. III, this indicates that different microscopic configurations appear with comparable, nonvanishing probability
@Eqs. ~3.1! and ~3.2!#. Such microscopic configurations,
when averaged with the Boltzmann probability distribution
~3.2!, lead to the diffusive interface. The relevant ~with nonvanishing probability! microscopic configurations in the
A. Ciach and R. Hołyst
3213
FIG. 7. The size d of the unit cell for the structures stable just below the
bifurcation line. The unit of length is the thickness of the surfactant monolayer. r s is the surfactant-volume fraction.
model studied here play a similar role as the undulations of
the sharp monolayer—they broaden the oil–water interface.
We have found that the effect of broadening of the interface
on the value S/ r s in different phases is different, and we
have a quantitative measure of its strength through the projected area. As expected, the effect is the stronger, the more
independent the surfactant-occupied surfaces are. Indeed, in
the lamellar phase the difference between the average and
the projected surface area is the largest, ~about 20% of the
relative difference!. In the C phase the structure is quite stiff
due to intersections of the surface, and there are fewer microscopic configurations deviating from the MF solution
coming from the minimization of Eq. ~3.5!. The effect of
broadening of the interface is weaker in the C phase than in
the other structures and consequently S/ r s is the largest as
shown in Fig. 6. In the special case of the C phase it is clear
from Eq. ~2.6! that the surface would intersect, therefore, in
the computation of S/ r s we have subtracted the volume occupied along the lines of intersection, since otherwise it
would be counted twice.
The value of Sd for the considered structures is known
~Sec. II!, and d is determined by the wave number corresponding to bifurcation, k b , so that for the L and P phases
d52 p /k b , for D and C phases d52 p )/k b and for G phase
d52 p &/k b ~see Fig. 7!. In our case the length unit is l and
should be identified with the length of the surfactant ~;25
Å!. From this estimate we find that d ranges from ;100 Å to
;200 Å.
The broadening of the surfactant profiles, due to the microscopic configurations39 taken into account via Eqs. ~3.1!
and ~3.2!, occur on the small length scale.39 It is set by the
size of the calculated f~r! profile. It gives at most 20% ‘‘absorption’’ of the surface area. On the other hand, the area
absorption in microemulsions in the typical system can be as
large as 80% due to the long wavelength undulations studied
by Helfrich40 for the lamellar phase. These fluctuations diverge logarithmically with the system size. In fact they can
be included in the model if we go beyond the mean-field
description of the Landau–Ginzburg model @Eq. ~3.3!#.
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3214
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
In concluding remarks we should mention limitations of
our approach. First, we only considered a selection of structures, and it is possible that some other, possibly more complicated structures stabilize as well and the phase diagram is
even more complicated and rich. The second limitation is the
bifurcation analysis, which can only be applied close to the
continuous transition between microemulsion and the cubic
phases. The full phase diagram of the model can only be
determined numerically in the future work. Finally, we use
the LG, mean-fieldlike approximation, in which fluctuations
are neglected. It is possible that the fluctuation-induced
weakly first-order phase transition can occur instead of the
continuous transition, as found in some models of such
systems.41–43 The effect of fluctuations in the simplified
version26 of the present model has been partially taken into
account in Ref. 43 and indeed, the fluctuation-induced firstorder transition was found. The D phase, found previously
within the bifurcation analysis, however, remained stable. It
is hence quite probable that in the present model the bicontinuous cubic phases remain stable beyond the LG approximation in the narrow window of surfactant volume fractions.
ACKNOWLEDGMENTS
We would like to thank Dr. Góźdź for many helpful
discussions. This work was partially supported by the KBN
Grant No. 2P03B12516 and the Maria Skłodowska Curie
Joint Fund II.
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