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From W. Góźdź, Bicontinuous Phases of Lyotropic Liquid Crystals. In: Aleš Iglič, Chandrashekhar V.
Kulkarni and Michael Rappolt, editors, Advances in Biomembranes and Lipid Self-Assembly, Vol. 23,
Burlington: Academic Press, 2016, pp. 145-168.
ISBN: 978-0-12-804715-6
© Copyright 2016 Elsevier Inc.
Academic Press
Author's personal copy
CHAPTER SIX
Bicontinuous Phases of Lyotropic
Liquid Crystals
W. Góźdź*,1
*
Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland
Corresponding author: e-mail address: wtg@ichf.edu.pl
1
Contents
1.
2.
3.
4.
Introduction
The Model and the Computational Method
Cubic Phases of Lyotropic Liquid Crystals
The Structure of the Interface Between the Cubic Phase and the Isotropic Phase
4.1 Computational Setup
4.2 The Structure of the Interfaces Between the Diamond and the Isotropic Phase
4.3 The Structure of the Interfaces Between the Gyroid and the Isotropic Phase
5. Conclusions
Acknowledgments
References
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Abstract
The bicontinuous lyotropic cubic phases and their properties are investigated within the
framework of the Landau–Brazovskii free energy functional. Cubic phases of different
symmetries and topology are examined. The structure and stability of the interfaces
between the cubic phases and the isotropic phases are described.
1. INTRODUCTION
Bicontinuous cubic lyotropic phases are liquids with crystalline order,
periodicity in x, y, and z directions having the length scale much larger than
the size of molecules. They are formed in systems with competing interactions [1–3], such as diblock copolymers, ternary mixtures of oil, water, and
surfactants, or binary mixtures of lipids and water [4–7], and they can be
encountered in biological cells in endoplasmic reticulum and organelle [8].
We focus on bicontinuous phases in mixtures of lipids and water.
The amphiphilic lipid molecules self-assemble into a bilayer with the hydrocarbon chains isolated from the water molecules. The bilayer divides the
Advances in Biomembranes and Lipid Self-Assembly, Volume 23
ISSN 2451-9634
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volume into two disjoint networks of channels filled with water and behaves
as an elastic membrane. For a fixed topology, the elastic energy of the membrane depends on the mean curvature, H, of the surface describing the center
of the bilayer according to the Helfrich formula [9]:
Z
Fb ¼ 2κ H 2 dA
(1)
where κ is the bending rigidity and dA is the infinitesimal area element.
The formula (1) may be applied for the systems with fixed topology
where according to Gauss–Bonnet theorem the surface integral of the
Gaussian curvature is constant. In equilibrium, the elastic energy assumes
the minimum. In the bulk, the minimum of (1) corresponds to triply
periodic minimal surface, because the minimal surfaces have zero mean
curvature at every point. The surface with zero mean curvature has saddlelike shape at every point (except flat points). A plane is a special case of the
minimal surface. The functional (1) has been successfully applied to the studies of vesicles built of lipid bilayers. The agreement between experimental
results and the calculations performed for different shapes of vesicles was
excellent [10–14].
The mathematical modeling of the bicontinuous phases in terms of
the functional (1) is however very difficult, especially in the presence of
the interface. It requires minimization of the functional (1) for functions
describing the location of the bilayer. These functions are not known a priori
and cannot be easily parameterized. Therefore, we decided to perform the
calculations within the framework of a relatively simple Landau–Brazovskiitype model with one scalar order parameter related to the local concentration of water [15–16].
2. THE MODEL AND THE COMPUTATIONAL METHOD
The free-energy functional for systems inhomogeneous on the
mesoscopic length scale can be written in the form
Z
F ½ϕðrÞ ¼ d 3 r ΔϕðrÞ2 + g½ϕðrÞrϕðrÞ2 + f ½ϕðrÞ :
(2)
This functional successfully describes systems as diverse as binary or ternary surfactant mixtures, block copolymers, colloidal particles with competing attractive and repulsive interactions of different range, or magnetic
systems with competing ferromagnetic and antiferromagnetic interactions
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[2, 3, 15, 17–20]. The order-parameter ϕ(r) in (2) depends on the physical
context. In the case of block copolymers or ternary water-surfactant-oil
mixtures ϕ(r) is the local concentration difference between the polar and
the nonpolar components. In the case of lipids in water ϕ2(r) describes
the local concentration of water. At the center of the lipid bilayer ϕ(r)¼0.
The bilayer divides the space into two disjoint water channels, one of them
on one side of the bilayer, where ϕ(r) > 0, and the other one on the other
side of the bilayer, where ϕ(r) < 0. The sign of ϕ(r) allows to distinguish
between the two disjoint channels of water. The fluid in each channel, however, has the same physical nature. For this reason in the case of lipids in
water the functional (2) must be an even function of ϕ.
In homogeneous phases, the concentration of water is independent of
the position. In such a case, ϕ(r)¼const and rϕ¼Δϕ¼0; therefore, f [ϕ]
is the free-energy density of the homogeneous phases. In the case of the
phase coexistence between water- and lipid-rich phases, we can postulate
for f [ϕ] the form with three minima of equal depth, where ϕ¼1 both represent the water-rich phase, f [ϕ(r)]¼(ϕ(r)2 1)2(ϕ(r))2. The inhomogeneous distribution of the components, in particular the formation of the
bilayer, is possible when the corresponding free energy is lower than for
any homogeneous structure. When the concentration ϕ(r) becomes position dependent, then (rϕ)2 >0. F can decrease for (rϕ)2 increasing from
zero when g[ϕ(r)]<0. On the other hand, the Laplacian term jΔϕj2 leads
to the increase of F , and the length scale of the inhomogeneities (the size
of the unit cell in the case of the periodic phases) is determined by the competition between the terms supporting and suppressing the spatial oscillations
of ϕ(r). Unlike in Refs. [2–3, 18–20] where g[ϕ(r)] was approximated by
a constant, we assume after Ref. [15] the form g½ϕðrÞ ¼ g2 ϕðrÞ2 + g0 with
g2 ¼4.01 g0 and g0 ¼ 3.
In the mean-field approximation, the stable or metastable phases of the
system correspond to the minimum of the functional (2). In order to minimize the functional, we discretize the field ϕ(r) on the cubic lattice [15].
The first and the second derivatives in the gradient and Laplacian term of
the Landau–Brazovskii functional at the point r ¼ (i, j,k)h on the lattice were
calculated according to the following formulas [21]
@ϕðrÞ ϕi + 1, j, k ϕi 1, j, k
,
!
2h
@x
and
(3)
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@ 2 ϕðrÞ
1
!
ϕi + 2, j, k + 16ϕi + 1, j, k
@x2
12h2
30ϕi, j, k + 16ϕi 1, j, k
ϕi 2, j, k ,
(4)
and similar in y and z directions.
The mixed derivatives are calculated [21] according to
@ 2 ϕðrÞ
!
@x@y
1
ϕ
+ ϕi 1, j, k + ϕi, j + 1, k + ϕi, j 1, k
2h2 i + 1, j, k
ϕi + 1, j + 1, k ϕi 1, j 1, k :
2ϕi, j, k
(5)
ϕi, j,k is the value of the field ϕ(r) at the point r ¼ (i, j,k)h, where h is the
lattice spacing. The calculation of derivatives on the lattice boundary
requires taking into account the points outside the lattice. These points
are given by appropriate boundary conditions.
After discretization, the problem of minimization of the functional is
converted to the problem of minimization of a function of many variables,
where the variables are the values of the field ϕ(r) at the lattice points. We
use the conjugate gradient method to minimize the function numerically
[15, 22, 23].
In order to limit the number of variables, the symmetry of the cubic
phases is exploited. Two images in Fig. 1 show the field inside a cube
(on the right side) and in the smallest tetrahedral element (on the left side)
obtained by the reduction of the volume inside the cube according to the
symmetries of the space group Im3m. The colors (different shades of gray
in the print version) represent the value of the field ϕ(r) at a given point
0.8
0.6
0.4
π
0.2
−0.0
−0.2
−0.4
−0.6
−0.8
Fig. 1 The field ϕ(r) inside the computational tetrahedron in a cubic unit cell. Green
(gray in the print version) and yellow (light gray in the print version) colors show
different sides of the surface ϕ(r)¼0.
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in space. The surface colored by green (gray in the print version) and yellow
(light gray in the print version) is given by ϕ(r) ¼ 0. The mapping of the field
values to the color is done according to the colormap presented in Fig. 1. By
exploiting the symmetry of the cubic phase, it is possible to reduce the volume of the cube 48 times. Thus, the number of the variables in the function
to be minimized can also be reduced.
In Fig. 2, we demonstrate how the cubic unit cell can be created from the
surface ϕ(r) ¼ 0 contained in the smallest tetrahedron and how the unit cell
can be moved by translating it to create the bulk bicontinuous phase. In the
first row, the computational tetrahedron with the surface ϕ(r) ¼ 0 (left side)
and the copies of this surface which form 1/8 of the cubic unit cell (right
side) are presented. The different copies of the surface from the smallest tetrahedral element are denoted by different colors (different shades of gray in
Fig. 2 Reconstructing the structure of the bicontinuous phase from the smallest tetrahedral element. Green (gray in the print version) and yellow (light gray in the print version) colors show different sides of the surface ϕ(r)¼0.
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the print version). In the second row, the unit cell and eight copies of the
unit cell which form a larger piece of the bicontinuous phase are presented.
In an analogous way, one can make calculations for cubic phases of other
symmetries, by the computational polyhedra having different shapes [15].
The order parameter field ϕ(r) carries enormous amount of information
about the local structure of the phases we have investigated. The most interesting is the topology of the phases, described by the surface ϕ(r)¼0. We use
a simplified version [15] of the marching cubes algorithm [23] to determine
the location of the surface ϕ(r)¼0. The surface ϕ(r)¼0 is given as a set of
connected triangles.
The triangulated surface is used to calculate the Euler characteristics, χ, of
the surface inside the computational cell. The calculation of χ can be done
according to the Euler formula χ ¼F + V E, where F, V, and E are the
number of faces (F), vertices (V), and edges (E) of the polygons cut out by
the surface ϕ(r)¼0 in the small cubes of dimension of lattice spacing. The
Euler characteristic for a closed surface is related to the Gaussian (K) curvature and the genus ( g) of this surface in the following way [24–25]
Z
1
χ¼
KdS ¼ 2ð1 gÞ,
(6)
2π S
where the integral is taken over the surface S. The genus is an integer number
and describes how many holes are in a closed surface. For example, the genus
for a sphere is zero, for a torus is one, and for a pretzel is two. The structures
we have investigated are infinite and periodic. The genus for an infinite surfaces is infinite, of course, but for a finite piece of this surface, in a unit cell, is
finite and characterizes the surface. Due to periodicity, the cuboidal cells can
be treated as closed surfaces in four dimensions, making the calculation of the
genus unambiguous [26]. Therefore, the genera of the structures inside the
cuboids can be calculated according to: g¼1 χ/2, where χ is the Euler characteristics for the surface inside the cubic computational cell. The Gaussian
and the mean curvatures are local characteristics of the internal surfaces given
by ϕ(r)¼0. In the description of the model, we have mentioned that some of
the structures in the model should be characterized by zero mean curvature at
every point of the internal interface. Here, we present the method used to
compute Gaussian and mean curvatures. The unit normal n(r) at the point
r is given by the gradient of the field ϕ(r) at the surface ϕ(r)¼0:
nðrÞ ¼
rϕðrÞ
:
rϕðrÞ
(7)
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The mean (H) curvature is given by the divergence of the unit vector
[27], normal to the surface at the point r, n(r)
2HðrÞ ¼ r nðrÞ
(8)
and the Gaussian curvature (K) by the formula [28–30]
2KðrÞ ¼ nðrÞ r2 nðrÞ + ½r nðrÞ2 + ½r nðrÞ2
(9)
In the numerical calculations of the curvatures, we used the following formulas [27, 31]:
H¼
1
B
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ϕ2x + ϕ2y + ϕ2z A
K¼
ϕ2x
(10)
1
C
2
2
+ ϕy + ϕz A
(11)
where A, B, and C are obtained from:
0
1
ϕxy
ϕxz
ϕx
ðϕxx λÞ
B ϕyx
ðϕyy λÞ
ϕyz
ϕy C
C ¼ Aλ2 + Bλ + C
det B
@ ϕzx
ϕzy
ðϕzz λÞ ϕz A
ϕx
ϕy
ϕz
0
(12)
and are given by:
A¼
ϕ2x + ϕ2y + ϕ2z
B ¼ ϕ2x ϕyy + ϕzz + ϕ2y ðϕxx + ϕzz Þ + ϕ2z ϕxx + ϕyy
2ϕx ϕy ϕxy
C ¼ ϕ2x ðϕ2yz
2ϕx ϕz ϕxz
ϕyy ϕzz Þ + ϕ2y ðϕ2xz
(13)
(14)
2ϕy ϕz ϕyz
ϕxx ϕzz Þ + ϕ2z ðϕ2xy
+ 2ϕx ϕz ðϕxz ϕyy
ϕxy ϕyz Þ + 2ϕx ϕy ðϕxy ϕzz
+ 2ϕy ϕz ðϕyz ϕxx
ϕxy ϕxz Þ
ϕxx ϕyy Þ
ϕxz ϕyz Þ
(15)
The mean and Gaussian curvatures have to be computed at the points of
the surface ϕ(r)¼0. The derivatives of the field ϕ(r) at the point r0, for
which ϕ(r0)¼0, are calculated according to the formulas (3)–(5).
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3. CUBIC PHASES OF LYOTROPIC LIQUID CRYSTALS
The most known lyotropic cubic bicontinuous phases are simple cubic
(P), gyroid (G), and double diamond (D). In Fig. 3, the results of the
minimization of the functional (2) are presented for three cubic phases with
different symmetry.
The length of the unit cell is different for each phase. The length of the
unit cell is related to the complexity of the internal structure of each phase.
The more complex the structure of the phase inside the unit cell is, the larger
the length of the unit cell becomes. It is interesting to note that the level
surfaces ϕ(r)¼0 closely resemble known triply periodic minimal surfaces
[32]. The level surface obtained from the P and D structures looks the same
as the primitive and diamond Schwarz minimal surface. The level surface
from the G structure is the gyroid surface discovered by A. Schoen. The
primitive, diamond, and gyroid minimal surfaces belong to the same family.
Each one of these three surfaces can be transformed into the other by the
Bonnet transformation.
The cubic phases presented in Fig. 3 are not the only minima of the functional (2). There are many bicontinuous structures which can be obtained by
minimization of the functional (2). In Fig. 4, we present such three structures
of simple topology and the same symmetry. They are known as IWP, BFY,
and CP structures [15], and also in this case there exist corresponding minimal surfaces.
In Figs. 5 and 6, the three structures of simple topology are presented,
known as the OCTO, CPD, and FRD. In the case of these surfaces, one
has to notice that the shape of the surface ϕ(r)¼0 can be quite different
Fig. 3 The simple cubic (P), gyroid (G), and double diamond (D) structures obtained by
the minimization of the functional (2). Green (gray in the print version) and yellow (light
gray in the print version) colors show different sides of the surface ϕ(r)¼0.
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Bicontinuous Phases of Lyotropic Liquid Crystals
Fig. 4 The IWP, BFY, and CP structures obtained by the minimization of the functional
(2). Green (gray in the print version) and yellow (light gray in the print version) colors
show different sides of the surface ϕ(r)¼0.
OCTD
CPD
Fig. 5 The OCTO and CPD structures obtained by the minimization of the functional (2).
Green (gray in the print version) and yellow (light gray in the print version) colors show
different sides of the surface ϕ(r)¼0.
depending on the location of the center of the cube of the unit cell. The
images on the left and the right side of each figure represent the same surface,
but the second image is taken when the center of the cube is in the origin of
the first one. Thus, one has to be careful when identifying these surface since
one may take the same surface for two different surfaces.
In the case of the FRD structure (Fig. 6), it is possible to choose the unit
cell which is not cubic. In Fig. 6, the third image shows the FRD structure
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Fig. 6 The FRD structure obtained by the minimization of the functional (2). Green (gray
in the print version) and yellow (light gray in the print version) colors show different
sides of the surface ϕ(r)¼0.
Fig. 7 The complex structures of different symmetry obtained by the minimization of
the functional (2). Green (gray in the print version) and yellow (light gray in the print
version) colors show different sides of the surface ϕ(r)¼0.
which is inside a rhombic dodecahedron instead of a cube. The faces of the
rhombic dodecahedron intersect the surface ϕ(r)¼0 in such a way that the
channels are perpendicular to the faces of the dodecahedron.
All the structures presented in Figs. 3–6 have been also obtained by the
minimization of the functionals which take into account only the shape of
the surface. In Fig. 7, we present the structures which were obtained only by
the minimization of the functional (2). The three complex structures shown
in Fig. 7 were obtained for three different symmetries of the cubic phases.
The symmetries of these structures correspond to the symmetries of the simple cubic, gyroid, and diamond phases, respectively (presented in Fig. 3). It
has to be noted that the complex structures were so far obtained only by the
minimization of the functional (2). In principle, these surfaces can also be
obtained by the minimization of an appropriate surface.
The complex structures from Fig. 7 and the structures presented in Figs.
8 and 9 were obtained only by the minimization of the functional (2). These
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Fig. 8 Simple cubic and multiply continuous simple cubic structures obtained by the
minimization of the functional (2). Green (gray in the print version) and yellow (light gray
in the print version) colors show different sides of the surface ϕ(r) ¼ 0.
Fig. 9 Gyroid and multiply continuous gyroid structures obtained by the minimization
of the functional (2). Green (gray in the print version) and yellow (light gray in the print
version) colors show different sides of the surface ϕ(r)¼0.
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structures are multiply continuous phases. They cannot be obtained just by
the minimization of a single surface or even a few surfaces. In Fig. 8, the
multiply continuous structures formed from the simple cubic phase are presented. The structures are drawn at the same scale. The length of the unit cell
depends on the number of surface ϕ(r)¼0 which is contained in the structure. We were able to obtain the structure with maximum four distinct surfaces. It seems that the number of the surface is limited, since when the
number of the surfaces increases they have to be more curved and thus the
energy of such more curved surface should increase. In Fig. 9, the multiply
continuous structures of the gyroid phase are presented. For the multiply
continuous gyroid phase, we have obtained the structures with three and five
distinct surfaces. Also in this case, the length of the unit cell increases when
larger number of distinct surfaces builds the structure of the multiply continuous phase.
4. THE STRUCTURE OF THE INTERFACE BETWEEN THE
CUBIC PHASE AND THE ISOTROPIC PHASE
In physical systems, triply periodic surfaces cannot extend to infinity.
They have to be bounded by the walls of a vessel or by the surrounding
coexisting phases. The latter case concerns, for example, the monocrystals
of cubic phases [33], or dispersion of nanoparticles built of cubic phases
in an isotropic phase [34–37]. When the bicontinuous phase coexists with
an isotropic phase, its structure at the interface has to be deformed in order to
avoid free edges of the bilayer exposed to the isotropic phase, because the
free edges are mainly built of hydrocarbon chains. Due to the hydrophobic
effect, the free edges merge with each other and the bilayer in the
bicontinuous phase forms a closed surface, where the surface is defined by
the midplane of the bilayer. It is obvious that it is not possible to close
the channels of the triply periodic surface with a zero mean curvature surface. The surface at the interface has to be built of the pieces of the surface
with positive and negative mean curvatures. This will result in an increase of
the bending energy (1). The bending energy increase depends on the direction of the interface with respect to the unit cell, since the periodic
bicontinuous phase is not isotropic. Unfortunately, the structure of the
interface between the bicontinuous and the isotropic phases cannot be
directly observed experimentally.
The knowledge about the structure of the surface of at the interface
could be obtained from appropriate theoretical calculations. So far,
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however, the theoretical description is restricted to purely geometrical
models based on the properties of triply periodic nodal surfaces [38] and
to lattice models [39–41]. The mathematical modeling based on the nodal
surfaces allows for investigation of topological properties of the studied
structures, but the shape of the bilayer and the elastic energy (1) of the examined object cannot be determined. Therefore, it is necessary to use appropriate free energy models to obtain the knowledge on the shape and the
excess energy of the bilayer at the interface with an isotropic phase.
4.1 Computational Setup
We shall minimize the functional (2) in the presence of the interface with the
water-rich phase for three different orientations of the unit cell of the cubic
bicontinuous phase. It is assumed that ϕ(r) is periodic in the directions parallel to the interface, while in the perpendicular direction the cubic
bicontinuous and the isotropic phases are present at the opposite sides of
the film. For each orientation an appropriate cuboidal cell is constructed.
We chose the coordinate frame with x and y directions parallel, and z direction normal to the surface of the interface [42]. The size of the cuboid in x
and y directions is determined by the value of the unit-cell length of the bulk
cubic bicontinuous phase and the orientation of the interface. The sizes of
the computational cells are different for each orientation due to different
symmetries in each case. The periodic boundary conditions in x and y directions and fixed boundary condition in z direction are applied, where the
field ϕ on one side of the cuboid is composed of the bulk cubic phase
and on the other side it is ϕ¼1 as in the isotropic phase.
We use the nodal approximation (Eqs. 16 and 17) to triply periodic
bicontinuous phases to build the initial configuration for the field ϕ(r) in
the cuboids:
2π
2π
2π
ψ D ðx1 , x2 , x3 Þ ¼ cos
x1 cos
x2 cos
x3
L
L
L
(16)
2π
2π
2π
sin
x1 sin
x2 sin
x3 ,
L
L
L
2π
2π
2π
2π
ψ G ðx1 ,x2 ,x3 Þ ¼ sin
x1 cos
x2 + sin
x3 cos
x1
L
L
L
L
2π
2π
+ sin
x2 cos
x3 ,
L
L
(17)
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where the coordinate frame is chosen such that xi are parallel to the edges of
the cubic unit cell of the bulk phase. The linear size L of the periodic cubic
element described by the nodal approximation (16) and (17) is taken as the
size of the unit cell of the bulk bicontinuous cubic phase, which is known
from previous calculations [15–16]. Such a cubic element of size L is shown
in Fig. 10B for the double diamond phase and in Fig. 11B for the gyroid
phase. We discretize the field ϕ(r) in such a way that we have 65 lattice
points on the length L. The initial configurations for all cuboids used to
determine the structure of the diamond and gyroid phase oriented in directions [111], [100], and [110] orthogonal to the interface are shown in Figs.
10A–C and 11A–C, respectively.
The size of the cuboids in x and y directions is equal to the value of
the period of the cubic phase in the corresponding direction. This size
is kept constant. In z direction the thicknesses of the cuboid are varied
in the range Zmin < Z < Zmax. Zmin is the size of the smallest cuboid in
z direction, equal to the period of the cubic phase in this direction and
Zmax ¼4Zmin.
A
B
C
Fig. 10 Computational cells used for different orientations of the diamond phase with
pffiffiffi
pffiffiffi
pffiffiffi
respect to the interface: (A) [111] 6=2L 2=2L 3L, (B) [100] L L L, and (C) [110]
pffiffiffi
pffiffiffi
2=2L L 2=2L. Green (gray in the print version) and yellow (light gray in the print
version) colors show different sides of the surface ϕ(r)¼0.
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Bicontinuous Phases of Lyotropic Liquid Crystals
A
B
C
Fig. 11 Computational cells used for different orientations of the gyroid phase with
pffiffiffi pffiffiffi pffiffiffi
respect to the interface: (A) [111] 6L 2L 3L, (B) [100] L L L, and (C) [110]
pffiffiffi
pffiffiffi
2L L 2L. Green (gray in the print version) and yellow (light gray in the print version) colors show different sides of the surface ϕ(r)¼0.
4.2 The Structure of the Interfaces Between the Diamond
and the Isotropic Phase
We calculated the structure of the interface perpendicular to three major
directions ([111], [100], [110]) by minimizing the functional (2) subject
to appropriate boundary conditions. Within the period of the cuboid (see
pffiffiffi
Fig. 10A) which is 3L along z axis, the interfaces can be formed at three
different positions when the channels of the bicontinuous phase are closed at
one side of the surface composed of amphiphilic molecules (see Fig. 12, left
column) and at three other positions when the channels of the bicontinuous
phase are closed on the other side of this surface (see Fig. 12, right column).
The distances between the positions of the subsequent interfaces formed at
different thickness of the bicontinuous phase are the same and equal to
pffiffiffi
3=6L. The formation of a new interface is related to the change of the
topology of the growing layer of the bicontinuous phase. The open channels
at the interface perpendicular to the [111] direction form a triangular lattice.
The surface of the interface is approximately flat. Such a unique structure of
this interface may explain their exceptional stability.
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Z
Fig. 12 The interface formed at different thickness of the bicontinuous phase. The new
interface is formed when the thickness of the bicontinuous phase is increased by
pffiffiffi
3=6L. Left and right panels show the interfaces with closed channels at one of the
two sides of the surface ϕ(r) ¼ 0. The surface of the first interface is bounded by the
dashed line.
The interfaces shown in Fig. 12 are obtained for different sizes of the
cuboids (reflecting the structures of different topology) for which the functional (2) has a minimum with respect to the thickness of the cubic phase in z
direction. The change of the topology is monitored by the calculations of
the genus of the surface ϕ(r)¼0 contained in the computational cell. The
genus is obtained by calculating the Euler characteristics of the surface
ϕ(r)¼0 [24–25, 27–31].
For the fixed topology (the same genus), we obtain configurations for the
interfaces perpendicular to the [111] direction for which the free energy is
optimized with respect to the thickness of the cubic phase in z direction. It
means that when the thickness is decreased or increased from the optimal
value, the free energy increases.
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Bicontinuous Phases of Lyotropic Liquid Crystals
In Fig. 13, we plot the values of the free energy for the interfaces
perpendicular to the [111] direction as a function of the thickness of the
cuboid F(Z). We also show the Helfrich free energy calculated for ϕ(r)¼
0 for the configurations obtained by minimization of the functional (2). It
is interesting to note that the positions of the minima are approximately
the same for the calculations performed for the functional (2) and the functional (1). Thus, we can speculate that the shapes of the facets obtained by
direct minimization of the functional (1) would be very similar to the shapes
obtained by direct minimization of the functional (2). It has to be noted that
minimization of the functional (1) is much more difficult since the parameterization of the surface of a cubic phase is practically impossible.
The interfaces perpendicular to the [100] direction can be formed at four
different positions within the period of the cuboid L in z direction. The subsequent interfaces are separated by the distance L/4. When a new interface is
formed, then the topology is changed. The shapes of the interfaces for the
thickness of the cubic phase in the range from z to z + L are presented in Fig.
14. Unlike in the case of the interfaces perpendicular to the [111] direction,
A
−0.1825
FL
−0.183
−0.1835
−0.184
−0.1845
5
B
4.5
FH
4
3.5
3
2.5
2
4
6
8
Z
Fig. 13 (A) The Landau–Brazovskii free energy per unit volume as a function of the
thickness of the cuboid (Z). (B) The Helfrich free energy per surface area of the plane
parallel to the interface as a function of the thickness of the cuboid (Z). The distance
pffiffiffi
on horizontal axis is given in units 3L, which is the period of the cuboid in z direction.
The circles and the squares denote the configurations with the open channels on one
or the other side of the surface ϕ(r)¼0.
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W. Góźdź
Z
Fig. 14 The interfaces perpendicular to the [100] direction formed at different thickness
of the bicontinous phase. The interfaces are formed when the thickness of the
bicontinuous phase is increased by 1/4L. Left and right panels show the interfaces with
closed channel at one of the two sides of the surface ϕ(r) ¼ 0. The surface of the first
interface is bounded by the dashed line.
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Bicontinuous Phases of Lyotropic Liquid Crystals
163
at the same thickness of the bicontinuous phase, two different interfaces may
be formed when the channels are closed on one or the other side of the surface
ϕ(r)¼0. These interfaces are structurally and energetically the same. The configuration in the left and right columns is obtained when the channels are
closed on green (gray in the print version) side or yellow (light gray in the
print version) side of the surface ϕ(r)¼0. At each row, the configurations
obtained for the same thickness of the cubic phase are presented.
The structure of the interfaces perpendicular to the [100] direction was
studied in a lattice model [39, 41]. In the continuous model, the order
parameter field is discretized on a large grid of points. Thus, the results of
calculations can describe the structure of the interfaces more accurately.
In the case of the interfaces perpendicular to [110] direction, two configurations with different topology can be distinguished within the period of
the cuboid along z axis. In Fig. 15, we show the configurations with the
Z
Fig. 15 The interfaces perpendicular to the direction [110] formed at different thickness
of the bicontinous phase. The interfaces are formed when the thickness of the
pffiffiffi
bicontinuous phase is increased by 2=4L. Left and right panels show the interfaces
with closed channel at one of the two sides of the surface ϕ(r) ¼ 0. The surface of
the first interface is bounded by the dashed line.
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W. Góźdź
lowest energy for the set of configuration with the same topology. The
interfaces perpendicular to the direction [110] are formed when the thickpffiffiffi
ness of the bicontinuous phase increases by 2=4L. The open channels of
the interfaces perpendicular to the direction [110] are arranged in a triangular
lattice for the optimal configurations. This is very similar to the configurations obtained for the interfaces perpendicular to the direction [111]. The
difference between these two types of interfaces is manifested when the
thickness of the cubic phase is not optimal. In the case of the interfaces perpendicular to the direction [111], the open channels are still arranged in a
triangular lattice, but for the interfaces perpendicular to the direction
[110] this triangular order is no longer present.
4.3 The Structure of the Interfaces Between the Gyroid
and the Isotropic Phase
From the calculations performed for the gyroid phase, it follows that the most
stable interface is for [110] direction. Within the period of the gyroid phase in
[110] direction, the interface can be created at four different values of the thickness of the gyroid phase. The structure of the gyroid phase for these unique
configurations is shown in Fig. 16. The left and right columns show the structure of the interface when the water channel of the gyroid phase is closed from
the first or the second side of the surface ϕ(r)¼0 which visualize the location of
the lipid bilayer. The configurations shown in Fig. 16 are obtained for different
sizes of the cuboids (reflecting the structures of different topology) for which
the functional (2) has a minimum with respect to the thickness of the cubic
phase in z direction. For the fixed topology (the same genus), we obtain configurations for which the free energy is optimized with respect to the thickness
of the cubic phase in z direction. It means that when the thickness is decreased
or increased from the optimal value, the free energy increases.
The distances between the positions of subsequent configurations
formed at different thickness of the gyroid phase are the same and equal
pffiffiffi
to 2=4L. The formation of a new interface is associated with the change
of the topology of the growing layer of the gyroid phase. The open water
channels in these configurations form a triangular lattice. The surface at the
boundary with the isotropic phase is approximately flat.
It is interesting to note that very similar behavior was observed for the
most stable orientations of the double diamond phase. The double diamond
phase was oriented in the [111] direction with respect to the surface of the
film. The open water channels also formed a triangular lattice.
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Bicontinuous Phases of Lyotropic Liquid Crystals
Fig. 16 The unique configurations of the structure of the interface between the gyroid
phase and the isotropic phase when the gyroid phase is arranged in the direction [110]
orthogonal to the interface. The new interface is formed when the thickness of the gyrpffiffiffi
oid phase is increased by 2=4L. Left and right panels show the configurations with
closed channel at one of the two sides of the surface ϕ(r) ¼ 0. Green (gray in the print
version) and yellow (light gray in the print version) colors show different sides of the
surface ϕ(r) ¼ 0.
[110]
[100]
[111]
Fig. 17 The unique configurations of the structure of the interface between the gyroid
phase and the isotropic phase when they are arranged in the direction [110], [100], and
[111] orthogonal to the interface. Green (gray in the print version) and yellow (light gray
in the print version) colors show different sides of the surface ϕ(r) ¼ 0.
In Fig. 17, we present the comparison of the interfaces between the gyroid and the isotropic phase oriented in three different directions. The most
stable is the interface oriented in the direction [110] and the least stable is the
interface oriented in the direction [111].
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W. Góźdź
5. CONCLUSIONS
We have used Landau–Brazovskii functional to investigate the cubic
bicontinuous phases of lyotropic liquid crystals. Many cubic phases of different topology and symmetry have been obtained. The structure of the interfaces between the cubic phases and the isotropic phase has been determined.
The stability of the interfaces has been analyzed by calculating also the bending energy per surface area for the surface ϕ(r)¼0 contained inside the computational cell. The configurations with the lowest energy are the same both
when the energy is calculated from the functional (2) where the field ϕ(r) is
taken into account and when the energy is calculated from the functional (1)
where only the shape of the surface ϕ(r)¼0 is considered. Thus, we can
safely restrict the analysis of the data obtained from minimization of the
functional (2) to the analysis of the properties of the surface ϕ(r)¼0. In
an ideal situation, the mean curvature of the ϕ(r)¼0 surface should be zero
at every point. In practice, in numerical calculations the mean curvature is
very close to zero. The surface ϕ(r)¼0 is deformed when the interface is
created. This deformation and thus deviation of the mean curvature from
the optimal value is the largest in the vicinity of the interfaces. When analyzing
the values of the local mean curvature on the surface ϕ(r)¼0, we observed
that the deformation of the surface due to formation of the interfaces propagates inside the cubic phase up to the distance of one, two unit cell length L.
We have estimated the surface energy associated with the formation of
the interface of the diamond and gyroid phase with the isotropic phase for
different orientation of the cubic phase with respect to the interface. We have
calculated that for the diamond and gyroid phases, the most stable orientation
of the interface is when it is perpendicular to the direction [111] and [110],
respectively. This is consistent with the results of the experiments on the soft
monocrystals and thin films of the lyotropic cubic phases [42–44].
ACKNOWLEDGMENTS
I would like to acknowledge the support from NCN grant No 2012/05/B/ST3/03302.
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