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Continuous Transformations of Cubic Minimal Surfaces Eur. Phys. J. B 7, 91–104 (1999) THE EUROPEAN PHYSICAL JOURNAL B EDP Sciences c Societ`a Italiana di Fisica Springer-Verlag 1999 Continuous transformations of cubic minimal surfaces A. Fogden1,a and S.T. Hyde2 1 Physical Chemistry 1, Center for Chemistry and Chemical Engineering, University of Lund, Box 124, 221 00 Lund, Sweden 2 Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Canberra ACT 0200, Australia Received: 5 March 1998 / Revised: 29 July 1998 / Accepted: 31 July 1998 Abstract. Although the primitive (P ), diamond (D)andgyroid(G) minimal surfaces form the structural basis for a multitude of self-assembling phases, such as the bicontinuous cubics, relatively little is known regarding their geometrical transformations, beyond the existence of the Bonnet isometry. Here their highest symmetry deformation modes, the rhombohedral and tetragonal distortions, are fully elucidated to provide a unified description of these simplest minimal surface families, with all quantities expressed in terms of complete elliptic integrals. The rhombohedral distortions of the gyroid are found to merge continuously with those which bridge the P and D surfaces, furnishing direct transformations between all three cubics, preserving both topology and zero mean curvature throughout. The tetragonal distortions behave analogously, offering an alternative route from the gyroid to the D surface. The cell axis ratios, surface areas and Gaussian curvature moments of all families are given, supplying the necessary geometrical input to a curvature energy description of cubic and intermediate phase stability. PACS. 61.30.-v Liquid crystals – 64.70.-p Specific phase transitions – 83.70.-f Material form 1 Introduction state is expressed as an expansion in the geometrical in- variants of the dividing surface. For a symmetric, sponta- The importance of crystalline bicontinuous morpholo- neously flat, bilayer the bending energy per unit midsur- gies in condensed atomic and molecular systems is now face area is then broadly recognized [1,2]. Examples can be found in co- g =2kH2+k¯K+... (1) valent atomic crystals, such as zeolites [2]; lyotropic c c liquid crystals [3–5], including lipid-water biochemical sys- in which H =(c1+c2)/2andK=c1c2 are the mean and tems [6,7] and synthetic surfactant systems [8]; meso- Gaussian curvatures, i.e. the first and second invariants porous inorganic materials synthesized in the presence of in the local principal curvatures ci. Moreover, if the bare amphiphiles [9–12]; in thermotropic liquid crystals [13,14]; repulsion of ionic surfactant bilayers across the solvent block copolymer melts [15,16]; ultrastructured biomin- domains is sufficiently short ranged, then its free energy eralized skeletons of sea-urchins [17], and cell mem- contribution for an arbitrary (weakly curved) interfacial branes [18], apparently in vivo. Related disordered bi- configuration can be incorporated consistently into this continuous morphologies are also prevalent in condensed general prescription, through an additional contribution ¯ systems, such as “sponge” mesophases in lyotropic sys- to the intrinsic bending moduli kc and kc [26]. Within tems [19–21], condensed mesoporous inorganics [22], and this approximation, the determination of free energies of late-stage spinodals [23]. Given the wide occurrence of bilayer systems reduces to the primarily geometrical task these morphologies, the need for a fundamental under- of choosing a sufficiently complete set of fluctuation modes standing of the range of competing structures, and their for summation. For a lamellar phase, the undulation am- attendant relative stabilities, is evident. plitudes are decomposed into Fourier modes and treated The subtlety of amphiphile aggregation behaviour, within the harmonic (or a pseudo-harmonic) approxima- revealed through structural studies of soft materials, tion. In the case of purely steric repulsion, the thermal continually challenges the existing framework of physi- fluctuations were found to give rise to a long ranged repul- cal theories for self-assembly. The classical DLVO the- sive force [25]. For charged lamellae, the transition from ory [24] of colloidal interactions requires a reformulation entropy- to energy-dominated repulsion with decreasing for microstructured fluids which accounts for the signif- ionic strength has been the subject of detailed investiga- icant thermal fluctuations of the interfaces. The basis tions [27]. for this renormalization is the bending Hamiltonian of In bicontinuous cubic (V2) and sponge phases the Helfrich [25]. The intrinsic free-energy of an interfacial bilayer partitions the solvent into a pair of interwoven multiply-connected tunnel systems. The Helfrich Hamil- a e-mail: [email protected] tonian implies that these structures, on average, be based 92 The European Physical Journal B upon midsurfaces which, as for the lamellar sheets, pos- mixtures [6–8,36]. Even more pertinent are the increas- sess zero mean-curvature, but now bear negative Gaussian ingly frequent observations of lyotropic liquid-crystalline curvature. Owing to the difficulty of geometrically formu- phases which display optically anisotropy. The vast major- lating an ensemble of hyperbolic interfacial configurations, ity of the known examples exhibit either rhombohedral or one cannot then aspire to the level of statistical mechani- tetragonal symmetry [6,37]. The resolution of interfacial cal sophistication attainable for lamellar phases. For ide- topology is subtle: these phases have, to date, been pre- alized disordered sponges some progress has, though, been sumed to be “mesh” structures (with 2-dimensional chan- possible using Gaussian random fields [28]. nel networks, in contrast to the 3-dimensional networks of For cubic phases, the focus of thermodynamic descrip- bicontinuous phases) [38]. The rhombohedral and tetrag- tions has rested principally on the zero temperature limit onal IPMS families explored here offer competing bicon- of a (fluid) bilayer midsurface possessing identically zero tinuous topologies of the same symmetry classes. mean-curvature throughout, i.e. an infinite periodic mini- Due to the general geometrical nature of this study, the mal surface (IPMS). Small angle scattering data collected results bear on the diversity of microstructured systems mentioned above. Detailed consideration of non-cubic bi- from V2 phases have to date [4–8] revealed three predomi- nant symmetries, corresponding to those of the three topo- continuous and mesh morphologies are equally applicable logically simplest examples of IPMS, the G (gyroid), D to these systems. For example, block copolymer melts ex- (diamond) and P (primitive) surfaces [29,30]. Although hibit liquid-crystalline mesophases analogous both in type these models have been broadly successful, their neglect of and sequence to the lyotropics. Even with transmission fluctuations gives rise to a ground-state degeneracy (van- electron microscopy, it is often difficult to clearly distin- ishing bulk and shear moduli) in the Helfrich description of guish bicontinuous and mesh phases since image textures cubic phases [31]. This degeneracy can, however, be lifted change drastically with the sample section [39]. The fam- by introducing higher orders to equation (1), specifically a ilies which link the simplest isotropic IPMS are likely fourth-order term in K2. To construct an ensemble of par- to offer a useful aid towards recognition of complex mi- titions about the three cubic surfaces, an obvious starting crodomain morphologies, and may help to resolve con- point is their lower symmetry generalizations, namely the troversies over G and D cubic structures in copolymer rhombohedral and tetragonal IPMS families obtained by systems [40]. distortion along a 3-fold or 4-fold axis, respectively. These special modes could be expected to furnish important con- tributions to fluctuation entropy, since it is the freedom of 2 Distortions of the D, G and P surfaces such minimality – preserving (H = 0) degradations which In Figure 1 we show the translational unit cells of the causes the harmonic degeneracy in the cubics. Our objec- ¯ ¯ tive in this paper is to explicitly construct these families of D, G and P surfaces, with space-groups Pn3m, Ia3d and Im3¯m. All three surfaces are “balanced” since they bear cubic distortions and determine their Gaussian curvature ¯ variations. embedded symmetry elements, i.e. 1 (inversion) points to- gether with 2-fold lines for D and P , which interchange Knowledge of the mathematics of lower symmetry the two sides of the surface and the two skeletal labyrinths IPMS has developed over the past few years [32–34], but is partitioned by it. Thus the 4-connected labyrinths of the D far from complete. Sadoc and Charvolin provided a topo- surface are related by translation, as are the 6-connected logical picture of the simplest pathway linking the G, D pair for the P surface, while the 3-connected networks and P surfaces, in which the junctions of their bicontin- for G are enantiomorphic. The overall cell topology is uous labyrinths are systematically merged [3]. A similar indexed by the Euler characteristic χ, which equals −2, mechanism, also phrased in terms of the labyrinth nets, −8and−4 per the units of D, G and P in Figure 1. If was recently discussed [35], with the argument that the the IPMS are used in an unbalanced context – e.g. the intermediate states of its cubic transitions would be ener- labyrinths are “colored” to be symmetrically distinct
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