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.
Back to home page Ulrich Schwarz.