Frustration in bilayers and topologies of liquid crystals
of amphiphilic molecules
J. F. Sadoc and J. Charvolin
Laboratoire de Physique des Solides, associé au CNRS, Bât. 510, Université Paris-Sud, 91405 Orsay, France
and GRECO microemulsions, CNRS, France
(Reçu le 22 juillet 1985, accepte le 5 dicembre 1985)
Résumé. - Les molécules amphiphiles construisent des films fluides symétriques, bicouches, le plus souvent organisés en phases lamellaires. Les comportements symétriques des deux couches d’une bicouche, particulière-
ment leurs changements de courbure avec les paramètres thermodynamiques, peuvent entrer en conflit avec la
compacité de la bicouche. Nous considérons le rôle possible d’une telle contrainte comme facteur de transfor- mation d’une phase lamellaire en phases cubique, hexagonale ou micellaire. Nous analysons cette contrainte
comme une frustration qui peut être résolue par l’introduction de disinclinaisons. Nous développons une approche géométrique et montrons que les solutions possibles sont topologiquement semblables à ces phases. Nous pro- posons donc de les décrire comme des structures de disinclinaisons. Cette approche géométrique met en évidence
le rôle structural des contraintes dans la bicouche et fournit le cadre dans lequel les autres facteurs physiques
doivent être considérés.
Abstract. - Amphiphilic molecules are well-known for their ability to build symmetric fluid films, or bilayers,
most often organized in lamellar phases. The symmetric changes of curvatures of the two interfaces of a bilayer,
as temperature and water content vary, may enter into conflict with the compactness of the bilayer core. We
consider the possible role of such a stress as a structural factor promoting the transformation of lamellar phases
into other liquid crystalline and micellar phases. We analyse this stress as a frustration which can be solved by
the introduction of defects of rotation, or disclinations. We develop a geometrical approach and show that the possible solutions correspond to organizations of amphiphilic molecules whose topologies are similar to those
of the liquid crystalline and micellar phases. We propose therefore to describe these phases as structures of discli- nations. Such a geometrical approach puts forward the structural role of internal stresses within bilayers and provides the frame within which the other physical factors are to be considered.
Classification
Physics Abstracts
61.30 - 02.40 - 82.70
1. Introduction.
We consider the particular case of molecules with a
strong amphiphilic character such as soaps, deter- gents or lipids. When these molecules are in the presence of water they aggregate so that their polar
heads remain in contact with water and most of the
methylene groups of their paraffinic chains are isolated
from the contact with water. The aggregates therefore present a paraffinic core limited by interfacial polar layers, as shown in figure 1. Structural studies of the
phases formed by such systems show a very rich
polymorphism which corresponds to different parti- tioning of space into amphiphilic and aqueous media with liquid-like behaviours [1-3]. The best known
structure is that of equidistant bilayers of amphiphilic
JOURNAL DE PHYSIQUE. - T. 47, N° 4, AVRIL, 1986
molecules with plane and parallel interfaces, the so-
called lamellar phase. Changes in the parameters of the phase diagrams, temperature and water content, promote transitions towards more complex ordered
or disordered organizations with curved amphiphile/
water interfaces, the so-called cubic, hexagonal and
micellar phases. The dominant factors of this poly- morphism are conjectured at the moment. We propose here that local properties of the bilayered organization play a dominant role in determining the topologies of
the structures and we explore the potentialities of a geometrical approach to this problem in the case
where the interfaces become convex on the water side.
In our point of view the results of such an approach provide the necessary framework within which the roles of other physical factors should be analysed.
46
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704068300
Fig. 1. - Centre : an aggregate with flat interfaces in a
lamellar phase (the dots represent the polar heads of the amphiphilic molecules and the short threads attached to the dots their paraffinic chains). The bilayer is made of two monolayers, each of them comprising a polar layer and a paraffinic layer. Left (right) : aggregates with curved inter- faces in hexagonal phases with water (paraffinic) cylinders
imbedded in a paraffinic (water) matrix.
As discussed in the second section « Local physical
factors and frustration » we consider that variations of the variables of the phase diagrams, water content and temperature, induce symmetric changes of the
curvatures of the two monolayers of a bilayer. These changes enter into conflict with the compactness of the paraffinic core. Such a situation can be described as
a frustration between curvature and thickness which
can be solved by changes of structure only. We
examine the possible solutions by developing a purely geometrical argument. For this we apply a method
used for solving similar cases of frustrations, as for
instance bi-dimensional tilings with regular polygons
and tri-dimensional packings of regular polyhedra [4],
which consists in finding first a curved space in which the structure is free of frustration, then defects of rotation are introduced according to the symmetries
of the structure in this curved space in order to map it on to the Euclidean space. This process generates
a structure without frustration in the Euclidean space but which differs from the initial one by the presence of defects. The application of this method requires abandonning the 3-D Euclidean space for a 3-D curved space. Facing the diffculty of pictorial repre- sentation in such spaces we think it useful to illustrate the method by discussing its application to a 2-D example in the third section « Geometrical solution in 2-D space ». The solutions of the frustration for a
bilayer in 3-D space are presented in the fourth section
« Geometrical solutions in 3-D space ». These solu- tions are then compared to the topologies of real
structures in the fifth section « Comparison with real
structures ». A satisfactory agreement is observed.
However one must not forget that it concerns the topology of the phases only. Other aspects of the polymorphism, symmetries of the structures and their sequence particularly, are dependent upon physical
terms which are not included in this approach.
Indeed the only physical terms considered here are
local, they concern the molecular states and inter- actions within a bilayer, and are responsible for the
frustration. Other physical terms, associated with
properties and interactions of fluid films, have to be
considered as well. Their possible role is briefly dis-
cussed in the last section «Comments on other
physical factors ».
2. Local physical factors and frustration.
Frustration is said to arise when two physical forces
oppose and lead to a compromise, i.e. neither of them
is fully satisfied. The impossibility for a bilayer to
curve its two monolayers without affecting the
compactness of its paraffinic core, as shown in figure 2, is a typical example of frustration. The state where the two monolayers are flat and parallel is not
frustrated as the compactness of the core is preserved.
States where the two monolayers are homogeneously
and symmetrically curved are frustrated as the compactness of the core is no longer preserved if the
uniaxial symmetry of the bilayer is maintained. The solution of such a frustration imposes a change of symmetry.
Fig. 2. - Centre : a bilayer with equal interfacial and mid
areas. Left (right) : the interfacial area is smaller (larger)
than the mid one.
The origin of frustration, or of the monolayer
curvature, has not been definitively modelized yet but the driving factors can be repertoried. In our point of view the mechanism can be found in the
independent behaviours of the polar and paraffinic layers constituting a monolayer [3, 5]. Their respective
areas do not necessarily vary in concordance when the parameters of the phase diagrams vary and difference of areas imposes a curving of the mono- layer. This point of view is supported by observation of structural changes induced in lamellar phases by
local perturbations changing the area of one layer of
the monolayer, the other one being kept constant [6, 7].
Specifically the structural changes induced by changes
of the chain length distribution in the paraffmic core,
while the polar heads and water content are kept unchanged [7], emphasize the fact that the paraffinic
medium is not passively submitted to the constraints
imposed by the hydrophoby and the electrostatics of the polar medium, but plays an active structural role. A second set of data provides some information about the nature of this role. They are NMR measu-
rements of the orientational disorder of the chain links in lamellar, hexagonal, nematic and micellar
structures. They show that the conformational state of the chains is quasi independent of the interfacial
curvature of the aggregate [8]. A mean field mode- lization of chains in aggregates of constant paraffinic
density has been developed recently to analyse these
data [9]. It shows that the conformational disorder of the chains is indeed nearly constant at a given temperature, close to that of an isolated random chain, whatever the shape of the aggregate. Thus compactness and entropy of the chains in the paraf-
finic medium appear as dominant terms controlling
the behaviour of this medium and the mean area per molecule in the paraffinic layer. They are different in nature from the interactions between polar heads
which control the behaviour of the polar layer and
the mean area per molecule on it These terms, and therefore the mean area per molecule in each layer
of a monolayer in figure 1, vary differently with temperature or water content. This means that, in the bilayer of a lamellar phase, the two interfacial areas
may become different from the mid area in the paraf-
finic core. This is a local stress which imposes a
structural change.
3. Geometrical solutions in 2-D space.
We deal with the section of a bilayer through one of
its normal planes and consider the case where the interfacial lengths become smaller than the mid ones, as shown in the centre and on the left of figure 2. The
frustration can be relaxed easily by wrapping the
section of the bilayer on a sphere with the mid line as a great circle, the equator, and the interfacial lines as
smaller equidistant circles, the parallels, as shown in figure 3. (This is always possible, whatever the dif-
I
Fig. 3. - The mapping of a frustrated bilayer in the flat Euclidean space R, onto a spherical space S2 where the
frustration is relaxed.
I’ )
ference of lengthy and the water content represented b the area of the sphere not occupied by the section
o the bilayer.) In doing so we have moved from a
re esentation of the section of the bilayer in the 2-D
e space R2 of its normal plane to its repre-
Euclidean space R2 of its normal plane to its repre-
Eu lidean space R2 of its normal plane to its repre- sen tion in the 2-D curved space S2 of the sphere.
To come back to R2, where the real structure is to be
found, it is necessary to decrease the curvature of S2.
This is to be done by introducing defects of rotation,
or disclinations, around the axes of symmetry of the relaxed structure in S2. They are one Coo axis and an infinity of C2 axes. Disclinations around the Coo
axis can be of any angle, their effect is simply to
increase the radius of the sphere, or to decrease its
curvature, without changing the thickness of the
Fig. 4. - A disclination around the Coo axis in S2.
bilayer, as shown in figure 4. In the limiting case of an
infinite angle of disclination the sphere S2 becomes locally equivalent to the plane R2. The result is a
section of bilayer with constant thickness but where the interfacial and mid lines are forced to have the
same length. Physically the energy involved in this
solution is that corresponding to the extension of the
polar layer. To introduce disclinations around one C2
axis the spherical space S2 is partly cut along the great circle of the mid line, an equator, and half a space is introduced in between the lips of the cut,
as shown in figure 5. Then, because of the tension of
Fig. 5. - A disclination around a C2 axis in S2.
the liquid films which imposes the mid lines to be geodesic curves on the surface, relaxation occurs
and two ( - n) disclinations are formed. This opera- tion is a step towards a solution as the thickness is
kept constant and the interfacial lines cannot but be smaller than the mid ones which are geodesic lines.
When an adequate density of disclinations is intro- duced the curved space S2 is mapped onto the flat
space R2 [10]. This is a solution topologically different
from the structure without frustration and from the first solution as it is made of an infinite number of closed cells. It is also physically different from the first solution since the interfacial and mid lengths
stay different but curved in some particular way.
This second solution shoud be preferred as curvature energies are generally smaller than stretching energies
in liquid crystals. There are several possible ways to introduce the disclinations in this second process.
Two extreme cases can be thought of where either
the same C2 axis is always used or each C2 axis is
used only once. They are represented in figure 6 at an
intermediate stage of the process. These two cases are
equally possible geometrically but not physically.,
Indeed the equilibrium laws of fluid films impose the
Fig. 6. - The disclinations are introduced either around the same C2 axis (left) or around different C2 axes (right).
solution on the right of figure 6 where the mid lines meet three by three at equal angles of 1200 [11].
At the end of the process a { 6.3 } hexagonal tiling [12]
of the flat space R2 is obtained if the same segment of additional space is introduced in each disclination.
In the case treated here, where the interfacial lengths
are smaller than the mid one as represented in the
centre and on the left of figure/2, we used a spherical
space with positive Gaussian curvature. We shall
not consider at this preliminary stage the opposite
case where the interfacial length become larger than
the mid one, as represented in centre and right of figure 2. By analogy with the previous case it is clear
that it can be treated in two ways : either by trans- ferring the bilayer of amphiphiles into a hyperbolic
space with negative Gaussian curvature or by trans- ferring a layer of water limited by two monolayers of amphiphiles into the same spherical space with
positive Gaussian curvature as above. Both situations,
which require supplementary introductory remarks
either about hyperbolic spaces or about interactions between facing interfaces through the aqueous layer,
will be analysed in details in future publications,
as well as their 3-D counter parts.
4. Geometrical solutions in 3-D space.
We now deal with the whole bilayer in the Euclidean space R3. It becomes frustrated when the two inter- facial areas become smaller than the mid area. This frustration is relaxed by moving into spherical space
83 which is an hypersphere with positive Gaussian
curvature. As the properties of the bilayer are sym-
metric relative to its mid surface, this mid surface must be a surface in 83 which divides it in two identical
subspaces. The way to look for such surfaces in S3
is also suggested by the above 2-D example. There
we chose the equator of sphere S2 as the mid line of the
section of the bilayer, since it is the shortest line which separates S2 into two identical hemispheres, and it is
at constant equal distances from the poles of the sphere, which are intersections of the sphere with
directions passing by the origin. This indicates that the search for surfaces Of S 3 which divide it in two identical
subspaces is that of the surfaces with stationnary areas equidistant from the intersections of 83 with directions
or planes passing by the origin. The equation of S3
C8.n be written either as X2 + X2 + X2 + X2 = R 2
or as x, = R cos 0 sin (p, X2 = R sin 0 sin T, X3 = R cos to cos cp, X4 = R sin to cos T. Its sections by one
direction x, = x2 = x3 = 0 are the points X4= ± R,
or (p = 0 and D = ± it/2. The great sphere S2, xf +
x3 = R 2 or co = 0, is equidistant from the two points
X4 = ± R as all its points are at an equal distance
± (n/2) R from them along geodesics. This sphere
is a first convenient surface. A second surface can be obtained considering the sections of S3 by the planes
x3 = x4 = 0 and Xt = X2 = 0. They are the great circles xi + X2 = R Z or cp = x/2 and X2 + x4 = R 2
or cp = 0. The surface corresponding to cp = n/4 is obviously equidistant from these two circles along geodesics associated with cp. Its equation is xi =
(R/,/2-) cos 0, X2 = (R/.,/2-) sin 0, X3 = (RIT2) cos m,
X4 = (R/,,/-2) sin co. It is called the spherical torus.
The infinitesimal length dl on the spherical torus is given by
this is the metrics of an Euclidean plane and, therefore,
the spherical torus has zero Gaussian curvature.
Indeed it can be built by identification of the opposite
sides of a square sheet of side J2 nR, as shown in
figure 7, and therefore it can be tiled by small identical
squares, a property which we shall use later on. There is no other surface with stationnary properties than
the great sphere and the spherical torus which divide S3 into two identical subspaces. Surfaces at equal
distances from the great sphere or from the spherical
torus are therefore identical and also have smaller
areas than them as the great sphere or the spherical
torus condense towards points or circles as the angular parameters Q) varies from 0 to ± n/2 or p varies from Tc/4 to 0 and n/2. A great sphere or a spherical torus can therefore be used to support a bilayer and relax its frustration in the curved space S3.
Fig. 7. - The transformation of a square into a spherical
torus by identification of its opposite sides in S3. Because
of the curvature of S3 this is done without elastic distorsion.
In the text we consider the two Coo symmetry axes (...) and
the family of C2 axes normal to the torus (-- ) only. We
have drawn also a second family of CZ axes on the surface (-)
which is generated by the diagonals of the square and their
parallels.
4.1 THE BILAYER ON A GREAT SPHERE OF 83. - In
this case the mid surface of the bilayer is the great sphere and the two interfaces are parallel spheres.
The symmetry axes of the sphere S2 in S3 can be
deduced by analogy from those of the plane in R3,
