Ruprecht Karls Universität Heidelberg

Bicontinuous cubic phases based on triply periodic minimal surfaces

In our recent work [SG99] we used a Ginzburg-Landau model for ternary amphiphilic systems with water-oil symmetry which has been used before to study eg statistical and dynamical properties of microemulsions [GS94]. It had been shown by Gompper and Zschocke [GZ92] as well as by Gozdz and Holyst [GH96] that this model can also be used to investigate bicontinuous cubic phases. The essential ingredient in our work is the Fourier approach, especially the implementation of black-and-white symmetries for the balanced cases [FK87] which correspond to the symmetry of the water and oil regions. The Fourier approach allows for efficient, systematical and easy-to-document numerics. A similar approach has been tried by Sheng and Elser [SE94]; however, the Ginzburg-Landau model they used describes phase separation where the interfaces are suppressed by positive surface tension, thus the lattice constant has to be fixed during minimization. In our case the Ginzburg-Landau model describes an ternary amphiphilic system with a stable lamellar phase. Both the interfaces and the lattice constant are stabilized by the interplay of a negative surface tension and positive bending rigidity. This leads to more stable numerics and to a simple criterion for the relative stability of the various single structures: their free energy density is to a very good approximation proportional to Gamma = Sqrt[A^3 / 2 pi chi a^6] where A, chi and a are surface area, Euler characteristic and lattice constant, respectively. We call this quantity topology index since it can be interpreted as a dimensionless Gaussian curvature (to the power of -1/2) independent of scaling and choice of unit cell. It is called homogeneity index by Hyde with a more mathematical motivation, since it can be interpreted to measure the deviation of variation of Gaussian curvature from a hypothetical homogeneous minimal surface with constant Gaussian curvature (for a nice recent work on Gamma see [FH99]). Note that since we deal with minimal surfaces, the dimensionless mean curvature Lambda = H a^3 / A (which we call curvature index) vanishes and Gamma is the only relevant quantity.

In the following table we collect data related to the seven balanced single structures investigated. The Ginzburg-Landau model has a three-dimensional parameter space and in order to compare with their results, for this table we use the point in parameter space chosen by Gozdz and Holyst [GH96].The structures are ordered with respect to increasing free energy densities f which is obtained by minimizing the Ginzburg-Landau functional numerically. chi and A are Euler characteristic and scaled surface area content for the conventional unit cell. For P, D and G, exact (Weierstrass) representations are known and the values for A are obtained by evaluating elliptical functions. For the other structures, we give our numerical results. In all cases, the volume fractions of the two labyrinths are both 0.5. Gamma is the topology index defined above. H and G are group and supergroup, respectively, which characterize the given balanced structure.

structure f chi A Gamma H G
G -0.19096 -8  3.09144  0.766668 I4132 Ia3d
S -0.18962 -40 5.41454 0.79474 I43d Ia3d
D -0.18870 -16  3.83779  0.749844 Fd3m Pn3m
P -0.18109 -4 2.34510 0.716346 Pm3m Im3m
C(Y) -0.18061 -24 4.46108  0.76730 P4332 I4132
C(D) -0.17382 -144 8.25578  0.78862 Fd3m Pn3m
C(P) -0.16239 -16 3.80938  0.74154 Pm3m Im3m

We also investigated two single non-balanced structures. Then there is no supergroup G and the volume fractions v and 1-v of the two labyrinths differ from 0.5. For I-WP, a Weierstrass representation and therefore the exact value of A is known.

structure f chi A Gamma v H
I-WP -0.18112 -12 3.46410 0.742515 0.536  Im3m
F-RD -0.16311 -40 4.75564  0.65417 0.532  Fm3m 

The difference of the Ginzburg-Landau results to the interface results for the complicated phases C(P), C(D), S, C(Y) and F-RD (that is the ones with a lot of modulation in the conventional unit cell and with large lattice constants) is larger than for the simple phases G, D, P and I-WP; this holds in particular for the topology index Gamma. When we exclude the complicated phases from our discussion, we find that the numerical results for the free energy density f and the topology index Gamma yield the same hierarchy. Thus the single gyroid G is expected to be the most stable of the various cubic bicontinuous phases in a ternary amphiphilic system at oil-water symmetry (equal amounts of oil and water and no spontaneous curvature). In our work we show how one can control the correction terms which result when mapping the Ginzburg-Landau model to the interface model and how these corrections can be made to vanish when using the GL-model's phase behavior in a clever way. This kind of reasoning which uses the GL-model and the interface model at the same time also explains why the single structures here are so close to minimal surfaces (essentially we solve the Willmore problem for triply periodic surfaces). One intriguing result it the high stability of the S-structure. We also believe that it has a high topology index Gamma, as indicated by our estimate for it and by the fact that it's skeletal graph features 3- and 4-coordinated vertices; note that G has 3-coordinated vertices and all other structures have at least 4-connected vertices. Note also that S is very similar to G (eg same supergroup G), thus it might well be that experimentalists have overlooked it's existence up to now.

By clicking on the specific structure' name above, you will get to a page with depicts it and gives the distribution of Gaussian curvature over the surface as a histogram which we got from your numerical results. There you also can download different files for three-dimensional viewing of the corresponding structures. OFF-files are offered for viewing with Geomview (my favorite graphics program; it is free software available from the Geometry Center - however not for Win95/NT).

Recently we used the data for the distribution of Gaussian curvature over the different cubic minimal surfaces for a detailed study of the stability of inverse bicontinuous cubic phases in lipid-water mixtures [SG00]. The midsurface of the lipid bilayer is minimal due to the the local symmetry between the two monolayers. For lattice constants large compared with the bilayer thickness, the monolayers can be assumed to be parallel surfaces to the minimal midsurface. In the framework of the curvature model, the free energy of each phase becomes a complicated function of the distribution of Gaussian curvature over the minimal midsurface. We showed that only P, D and G are stable since they share the same narrow distribution of Gaussian curvature due to a Bonnet symmetry. This also leads to the result that these three phases coexist along a triple line; the sequence G, D and P with increasing water concentration is determined by the geometry index. In fact the P-phase turns out to coexist with an excess water phase: due to spontaneous curvature for the monolayers, the system prefers to keep the favorable lattice constant for the ordered phase and expels any additional water. This is known as emulsification failure from surfactant systems, and now explains nicely how biological bilayers can exist in excess water.

Selected Related Links

  • The Scientific Graphics Project headed by David Hoffman offers many stunning pictures of minimal surfaces obtained from their exact (Weierstrass) representations. They also offer the software free of charge which they developed to generate the pictures.
  • The Center for Geometry, Analysis, Numerics & Graphics (GANG) is an interdisciplinary Differential Geometry research team in the Dept of Mathematics & Statistics at the University of Massachusetts, Amherst, Massachusetts, USA. It offers great pictures for surfaces of constant mean curvature (CMC), of which minimal surfaces are a special case.
  • Ken Brakke worked on TPMS with the Surface Evolver, a program which he also gives away free of charge. This is a finite element method where the surfaces are represented by an ensemble of triangles and the surface area (which is the sum of the many triangles' surface areas) is minimized for fixed topology and lattice constant. In fact this program works very well and this approach is often adopted by mathematicians. Besides, there are many more physical applications to it (like soap films, foams, wetting, etc.), and many physicists working on amphiphilic systems also use this program for generating TPMS and CMC-surfaces.
  • Karsten Grosse-Brauckmann used the Surface Evolver to investigate the single gyroid structure and the family of triply periodic surfaces of constant mean curvature which belongs to it [KGB97].
  • Konrad Polthier (esp. together with Karcher) discovered many new TPMS and existence theorems during the last decade [KP96].
  • Antonio Ros from the University of Granada, Spain, offers a great website with original research papers and introductory material on minimal surfaces and isoperimetric problems. Moreover he is involved with the surface group of Granada University, which offers a nice collection of minimal surface related material.
  • Wojciech Gozdz and Robert Holyst were the first to discover that the Ginzburg-Landau model we use has local minima which are close to TPMS (their claim that they are minimal surfaces however was somehow to strong) [GH96]. One difference to our work is that they worked with real space minimization. They also used the Ginzburg-Landau model to investigate multicontinuous and high genus structures.
  • Stephen Hyde has done a lot of work in this direction, mostly working with the exact Weierstrass representations (see eg [FH99]).
  • Reinhard Nesper from the ETHZ together with von Schnering developed the concept of nodal surfaces for TPMS [SN91]. Here they are also represented as isosurfaces to scalar fields. However, only as little Fourier modes are used as are necessary to represent the topology of a certain structure considered correctly. These approximations are widely used and quite easy to work with. In our work we give improved nodal approximations, that is we add some more Fourier modes in order to improve on the curvature properties of these representations.
  • Humberto Terrones suggested together with Mackay that structures close to TPMS might also occur with carbon systems [MT91]. These structures are called Schwarzites. In constrast to fullerenes (which only feature 6- and 5-sided polygons), their negative Gaussian curvature must lead to 7- and more sided polygons. However, up to now no synthetic pathway has been found.
  • The Liquid Crystals Group at the Department of Chemistry, Imperial College of Science, London, is headed by Dr John Seddon and Dr Richard Templer, who are experts on lipid polymorphism. In particular, they measured the phase diagram for 2:1 lauric acid/dilauroyl phosphatidylcholine and water, which is the only phase diagram I know of in which all three phases G, D and P are stable [TS98]. Indeed, this experimental phase diagram agrees very well with our theoretical prediciton for the generic case [SG00].
  • Martin Caffrey from Ohio State systematically surveys lipid phase behavior in order to estimate its use for bio-engineering purposes, eg for crystallization of membrane proteins. In particular, he measured the equilibrium phase diagram of monoolein and water [QC00]. The bicontinuous cubic phases occuring in this system have recently been used to crystallize bacteriorhodopsin for the first time in a three-dimensional structure [PP97]. In fact it also agrees well with our theoretical phase diagram [SG00], except that P is not stable in this system.
  • Ed Thomas from MIT worked for many years on structure determination in diblock copolymer systems.

Selected References

[SG00] U. S. Schwarz and G. Gompper. Stability of inverse bicontinuous cubic phases in lipid-water mixtures. Phys. Rev. Lett. 85: 1472-1475 (2000). U. S. Schwarz and G. Gompper. Bending frustration of lipid-water mesophases based on cubic minimal surfaces. Langmuir 17: 2084 -2096 (2001).
[SG99] U. S. Schwarz and G. Gompper. Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems. Phys. Rev. E 59: 5528 - 5541 (1999).
[GS94] G. Gompper and M. Schick. Self-assembling amphiphilic systems. Vol. 16 of Phase transitions and critical phenomena, eds. C. Domb and J. L. Lebowitz. Academic Press, London (1994).
[GZ92] G. Gompper and S. Zschocke. Ginzburg-Landau theory of oil-water-surfactant mixtures. Phys. Rev. A 46: 4836 - 4851 (1992).
[GH96] W. Gozdz and R. Holyst. Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. Phys. Rev. E 54: 1-16 (1996).
[FK87] W. Fischer and E. Koch. On 3-periodic minimal surfaces. Z. Kristallogr. 179: 31-52 (1987). See also W. Fischer and E. Koch. Spanning minimal surfaces. Phil. Trans. R. Soc. Lond. A 354: 2105-2142 (1996).
[SE94] Q. Sheng. Minimal surfaces with prescribed space group symmetry. PhD-thesis Cornell University 1994, supervised by V. Elser.
[KGB97] K. Grosse-Brauckmann. On gyroid interfaces. J. Colloid Interface Sci. 187: 418-428 (1997).
[FH99] A. Fodgen and S. T. Hyde. Continuous transformations of cubic minimal surfaces. Eur. Phys. J. B 7: 91-104 (1999).
[KP96] H. Karcher and K. Polthier. Construction of triply periodic minimal surfaces. Phil. Trans. R. Soc. Lond. A 354: 2077-2104 (1996).
[SN91] H. G. von Schnering and R. Nesper. Nodal surfaces of Fourier series: fundamental invariants of structured matter. Z. Phys. B 83: 407-412 (1991).
[MT91] A. L. Mackay and H. Terrones. Diamond from graphite. Nature 352: 762 (1991).
[TS98] R. H. Templer, J. M. Seddon et al. Inverse bicontinuous cubic phases in 2:1 fatty acid/phosphatidylcholine mixtures. The effect of chain length, hydration, and temperature. J. Phys. Chem. B 102: 7251 (1998).
[QC00] H. Qiu and M. Caffrey. The phase diagram of the monoolein/water system: metastability and equilibrium aspects. Biomaterials 21: 223 (2000).
[PP97] E. Pebay-Peyroula et al. X-ray structure of bacteriorhodopsin at 2.5 angstroms from the microcrystals grown in lipidic cubic phases. Science 277: 1676 (1997).

Mathematica file for TPMS

Here is a mathematica file which offers most of the the nodal approximations given in [SG99] plus the improved nodal approximations for P, D, G and I-WP given in [SG99]. It can be used for obtaining visualizations of approximations to TPMS and as a starting point for physical studies which need representations of cubic bicontinuous structures. The program also allows to measure mean and Gaussian curvature for a triangularization of all nodal surfaces.

Last modified Mon Oct 29 11:15:20 CET 2001.
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