Bicontinuous Cubic Phases in Amphiphilic Systems
Introduction
Amphiphilic molecules have both polar and non-polar parts (eg a
polar headgroup connected to a hydrocarbon chain). Important examples are
tensides and lipids. We speak of binary amphiphilic systems if the
amphiphile is mixed with a polar or a non-polar solvent like water or oil;
and of ternary amphiphilic systems if the amphiphile is mixed with
both a polar and a non-polar solvent. Due to the hydrophobic effect, the
amphiphiles self-assemble into aggregates. In binary amphiphilic systems
the basic aggregates are (spherical or cylindrical) micelles and bilayers;
in ternary amphiphilic systems, monolayers form at the oil-water interfaces.
Depending on temperature, concentration and salt content, these aggregates
form many different phases each of which corresponds to a specific arrangement
of the amphiphilic aggregates' neutral surfaces. One of the intriguing
aspects of amphiphilic polymorphism is the existence of bicontinuous phases
which can be traversed in any direction in both the hydrophilic (water-like)
and the hydophobic (oil-like) regions. This bicontinuity has been
demonstrated by measuring diffusion properties with nuclear magnetic resonance.
In contrast to sponge and microemulsion phases, which are bicontinuous
as well, bicontinuous cubic phases show long range order which can
be demonstrated by the appearance of Bragg-peaks in diffraction patterns.
Although bicontinuous ordered phases could have any of the 230 three-dimensional
space groups, experiments show that most of them have one of the 36 cubic
ones. The amphiphilic interfaces of bicontinuous cubic phases form triply
periodic surfaces (TPS). Triply periodic means having a three-dimensional
Bravais lattice. It is easy to see that two- or one-dimensional Bravais
lattices (like with the doubly periodic hexagonal phase or the singly periodic
lamellar phase, respectively) cannot correspond to structures which are
continuous both in their hydrophilic and hydrophobic regions.
A TPS divides space into two unconnected but intertwined labyrinths.
Both labyrinths percolate space and provide the pathways which can be used
to traverse the structure both in the hydrophilic and the hydrophobic regions.
In most relevant cases, the two labyrinths are congruent to each other.
Then the structure is called balanced. A balanced structure is characterized
by two (rather than by one) space groups: the space group H of a single
labyrinths and the space group G of both labyrinths together. H can be
also be considered to be the space group of the oriented TPS as
opposed to G, the space group of the unoriented TPS. H is
a subgroup of index 2 of G since it comprises all symmetry operations of
G expect the one operation alpha which interchanges the two labyrinths
(i.e. maps one side of the TPS onto the other side). G is a supergroup
of H since it comprises all symmetry operations of H plus alpha. Therefore
G = H x Z_2 with Z_2 = {1, alpha} being the cyclic group of order 2 which
has as its members the unit operation and alpha.
Ternary amphiphilic systems
In ternary amphiphilic systems with water, oil and amphiphile, water and
oil regions are separated by amphiphilic monolayers and bicontinuous cubic
phases can come in three structural types:
-
Single structure: there is only one TPS covered by an amphiphilic
monolayer and the two labyrinths are filled with water and oil, respectively.
-
Double structure of typ I: the two labyrinths of the single structure
are filled with oil. Thus the TPS of the single structure is covered by
a water-filled (reversed) bilayer, i.e. a sheet-like water region
enclosed by two amphiphilic monolayers.
-
Double structures of typ II: this is the case before with water
and oil interchanged. Thus the TPS of the single structure is covered by
an oil-filled (normal) bilayer, i.e. a sheet-like oil region enclosed
by two amphiphilic monolayers.
Since one picture says more than thousands words, look at the following
picture which depicts the essential elements of the so-called P-structures:
On the left you see the two labyrinths in green and red, respectively.
They are congruent to another, and the operation which maps one onto the
other is a translation by half the unit cell's body diagonal. In each vertex,
six channels meet each other perpendicularly. Each labyrinth has the space
group H = Pm3m (simple cubic). If we now plug one labyrinth into the other
and identify green and red, we get the new structure on the right which
has space group G = Im3m (body centered cubic). The dividing surface
is triply periodic and shown in blue. Note that by construction, the translation
by half the unit cell's body diagonal not only interchanges the two labyrinths,
but also appears to map the TPS onto itself. Thus the surface has space
group G = Im3m also. However, this is not totally correct, since at the
same time the two sides of the surfaces are interchanged. As explained
above, this means that only the unoriented surface has space group G =
Im3m; if we consider the TPS to have two sides (e.g. light and dark blue
or green and red or white and black), this operation is not a symmetry
operation and the space group is H = Pm3m only. (In fact, this kind of
additional symmetry is called a color symmetry and constructions
like this are also known from other part of condensed matter physics like
magnetic crystals, Fermi surfaces and zero potential surfaces in ionic
crystals). Now how do the P-structures look like in a physical system like
the ternary amphiphilic system ? For the single P, the green labyrinth
might be filled with oil and the red one with water. The blue surface is
an amphiphilic monolayer. For double structures of typ I and II both labyrinth
are filled with the same component, i.e. oil and water, respectively. Then
the TPS is covered by two monolayers which sandwich the water and oil region,
respectively.
Binary amphiphilic systems
If we remove the oil, the ternary system becomes a binary amphiphlic system
with water and amphiphile. However, there are still three different structural
types possible:
-
Single structure: one network of cylindrical micelles with space
group H in a water matrix.
-
Double structure of typ I: two networks of cylindrical micelles
with space group G in a water matrix.
-
Double structures of typ II: two networks of water channels with
space group G separated by an amphiphilic bilayer. Or expressed differently:
two networks of inverse cylindrical micelles filled with water (thus these
phases are often called inverse cubic phases).
Due to their biological relevance, inverse cubic phases are well investigated
with lipid-water mixtures (work by Luzzati, Fontell, Seddon, Templer, Hyde,
Anderson, etc). The most common one corresponds to the G-surface, but D-
and P-surfaces are established also. The sponge phase can be considered
to be a molten variant of the inverse cubic phases, although of a lattice
constant much larger than one normally finds for inverse cubic phases.
For binary double structures of typ I, only the G-surface is confirmed
(for surfactant-water systems). I do not know any experimental account
of binary single structures. For ternary systems, much less experimental
results are known. One exception is the system DDAB-water-hydrophobe, which
was investigated by several authors. Stroem and Anderson report that here
five different cubic phases exist, all of them being ternary double structures
of typ II. Although many ternary amphiphilic systems feature cubic bicontinuous
phases, indexing seems to be much more difficult here than with binary
amphiphilic systems. Since the ubiquitous microemulsion consists of amphiphilic
monolayers separating continuous regions of water and oil, one expects
to find ternary single structures also. However, except for one or two
papers (e.g. by Raedler et al.), not much has been reported by experimentalists
along these lines. On the other hand, in the Ginzburg-Landau models I worked
with, the simple structures seem to be quite favorable both for binary
and ternary systems. A stable single structure was also found for a lattice
gas model for binary systems by Linhananta and Sullivan. Therefore it well
might be that the experimentalists have not looked closely enough yet.
This stands in marked contrast to the fact that there is a very nice experimental
way to get a characteristic signature of balanced single structures: by
deuterating the components, one can switch from bulk to film contrast and
thus from measuring spacegroup H to measuring spacegroup G in small angle
neutron scattering (doing the same for a double structure should not yield
any change in space group).
Bicontinuous cubic phases based
on triply periodic minimal surfaces
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