Provided for non-commercial research and educational use only. Not for reproduction, distribution or commercial use. This chapter was originally published in the book Advances in Biomembranes and Lipid Self-Assembly, Vol. 23 published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues who know you, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier's permissions site at: http://www.elsevier.com/locate/permissionusematerial 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 145 146 152 156 157 159 164 166 166 166 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 http://dx.doi.org/10.1016/bs.abl.2015.12.003 # 2016 Elsevier Inc. All rights reserved. 145 Author's personal copy 146 W. Góźdź 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 ½ϕðrފ ¼ d 3 r ΔϕðrÞ2 + g½ϕðrފrϕðrÞ2 + f ½ϕðrފ : (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 Author's personal copy Bicontinuous Phases of Lyotropic Liquid Crystals 147 [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 g½ϕðrފ 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) Author's personal copy 148 W. Góźdź @ 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. Author's personal copy Bicontinuous Phases of Lyotropic Liquid Crystals 149 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. Author's personal copy 150 W. Góźdź 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) Author's personal copy 151 Bicontinuous Phases of Lyotropic Liquid Crystals 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
nðrފ2 + ½r  nðrފ2 (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). Author's personal copy 152 W. Góźdź 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. Author's personal copy 153 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 Author's personal copy 154 W. Góźdź 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 Author's personal copy Bicontinuous Phases of Lyotropic Liquid Crystals 155 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. Author's personal copy 156 W. Góźdź 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, Author's personal copy Bicontinuous Phases of Lyotropic Liquid Crystals 157 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) Author's personal copy 158 W. Góźdź 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. Author's personal copy 159 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. Author's personal copy 160 W. Góźdź 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. Author's personal copy 161 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. Author's personal copy 162 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. Author's personal copy 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. Author's personal copy 164 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. Author's personal copy 165 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]. Author's personal copy 166 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. 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