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Textures of S3 blue phase
B. Pansu, E. Dubois-Violette
To cite this version:
B. Pansu, E. Dubois-Violette. Textures of S3 blue phase. Journal de Physique, 1987, 48 (11), pp.1861- 1869. �10.1051/jphys:0198700480110186100�. �jpa-00210628�
Textures of S3 blue phase
B. Pansu and E. Dubois-Violette
Laboratoire de Physique des Solides, Université Paris-Sud, Bât. 510, Centre d’Orsay, 91405 Orsay Cedex,
France
(Requ le 9 mars 1987, accept6 le 8 juillet 1987)
Résumé. 2014 Nous décrivons des configurations non singulieres de phase bleue sur S3. Ces textures sont
engendrées par des champs de vecteurs sur S3. Les propriétés topologiques de la sphère S3 entraînent
l’existence d’un grand nombre de textures appartenant à différentes classes d’homotopie. Nous avons construit
certains champs de directeur satisfaisant une condition locale de minimum d’énergie. Pour faire cette
construction, nous avons utilisé les propriétés d’applications harmoniques. Nous donnons une description topologique des champs ainsi obtenus.
Abstract. - We describe non singular configurations of S3 blue phase. These textures are generated by vector
fields on S3. The topological properties of the sphere S3 are such that there exist numerous textures belonging
to different homotopy classes. We have constructed some director fields satisfying a local condition of energy minimum. This is done with use of some harmonic maps. We analyse the topology of the resulting director field lines.
Classification
Physics Abstracts
61.30 - 02.40
1. Introduction.
Blue phases of liquid crystal have been described locally by a double twist of the director n [1]. In
cholesteric phases the director twists in one direction
perpendicular to n ; there exists perfect cholesteric
phases in (R3. As indicated by several authors [2, 3],
the double twist condition implies frustration in
(R3 ; perfect blue phases cannot exist in R 3. On the other hand, it has been shown [2] that the double twist condition can be realized everywhere on a 3-
dimensional sphere S3. In a more general way we introduced in a preceding paper the construction of a double twist connection on any Riemannian 3-man- ifold [3]. We also showed that the S3 blue phase is a
nematic phase in the Lie algebra of S3. We described
a perfect configuration of blue phase with use of a special fibration of S3. In this paper we show that
non singular configurations or textures of blue phases are numerous which is not the case for nematic textures [4, 5] in R3. From the mathematical
standpoint a texture is defined by a non singular
director field n (P ). As indicated in section 2, search- ing non singular director field (of nematic or blue phases) is equivalent to finding non singular vector
fields. Nematic or blue phase textures are then
described by mappings f: R 3 __+ S2 or S3 -+ S2. These
maps or textures can be classified from a topological point of view : equivalent textures will belong to the
same homotopy class. For nematics there exists only
one homotopy class of maps 1R3 -+ S2 : all textures are topologically equivalent. This is no longer the
case if one considers particular boundary conditions [6, 7]. On the other hand textures of blue phase are specified by mappings f: S3 _+ S2 classified by
7T3(82) = Z. Numerous different classes of textures
are then possible. Within each class some textures do correspond to a local minimum of the elastic free energy. The director field satisfies the local equation
of minimum energy :
In both cases, nematic in R3 or S3 blue phase, A is
the Laplace-Beltrami operator acting on o-forms i.e.
functions on R3 or S3. Solutions of equation (1) for points P E S3 define harmonics maps S3 -+ S2. In section 3 we use this property to construct some of the Z possible textures and give the explicit ex- pression of the director field. Textures corresponding
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110186100
1862
to maps f: S3 --+ S2 are classified, in section 4, by an integer, the Hopf invariant H(f). It defines the
homotopy class to which the maps belong to. We analyse the related textures and show that they verify the harmonicity condition (1). We find that a
field belonging to a class of order p2 has an energy
proportional to p. We also make explicit characteris- tic features of the director field topology.
2. Textures.
2.1 DIRECTOR FIELD. VECTOR FIELD. - Let us call texture a non singular configuration of liquid crystal
molecules generated by a non singular director field
n(P). The order parameter, the director, is defined by a unit vector n (P ) with the condition that
+ n (P ) and - n (P ) are equivalent. The space of the order parameter is then the projective plane RP2. A texture, corresponding to nematic samples where no boundary conditions are imposed, is then described
by a map f : 1R3 --+ RP2. (One associates to each point
P of R 3 a value of the director n(P ) i.e. a point of Rp2.) The classification of textures depends strongly
on the boundary conditions. For example, if one imposes a uniformly fixed orientation of the molecules at infinity (7), the maps f extend to
S3 (R3 U oo ) --+ Rp2 and are therefore classified up
to homotopy by 7T3(IRP2) = Z.
In what follows we are essentially concerned with textures of S3 blue phase (the physical space is
S3) which are described a map f :
The order parameter space IRP2 is not simply connec-
ted. This is easily seen in a representation where the
order parameter space is a sphere S2 with identifica- tion of antipodal points (equivalence of + n and
- n). From a mathematical standpoint the first homotopy group 7Tl (IRP2) = 7L2 reveals this non
simply connectedness of the order parameter space.
From a general point of view [8] a map f from one manifold X to a manifold Y.
can be lifted to a map f :
on universal covering spaces (simply connected manifolds) denoted X, Y. The universal covering
spaces X and Y are fiber bundles with bases X and Y and fibers 7Tl (X) and 7Tl (Y). The covering space of the order parameter space Rp2 is S2. For nematics the physical space X is (R3 then X - X. The map
f which defines the texture is :
There is only one class of such maps : all textures are
topologically equivalent. For the S3 blue phase the manifold X = S3 and 7Tl (S3) = 0, which implies X - X = S3. . Then here again every non singular
director field on S3 is a vector field on S3. Let us
emphasize that this is no longer the case in presence of defects. Indeed in that case the manifold X is the total space in which the defect points are excluded.
It results that now w 1 (R 3 - defects) o 0 and 7Tl (S3 - defects) :0 0 and then director fields are no
longer necessarily vector fields.
Textures of S3 blue phase generated by maps
f: S3 --+ S2 are therefore classified by 7T3 (S2) = Z.
Each homotopy class is defined by an element of
1T 3(S2). Our purpose is now to select among all the Z
possible textures the ones which do correspond to
stable structures.
2.2 LOCAL MINIMUM OF THE ENERGY. HARMONIC
MAPS. - Metastable textures correspond to a local
minimum of the energy F. We shall now give the
exact condition of local minimum of the energy and show that for the S3 blue phase it defines harmonic maps f : S3 -+ f S2. We first recall the situation in nematic phases and then show it leads to a similar
expression for the S3 blue phase.
- Nematic phase. - The Frank elastic free ener-
gy simply reads :
where we use the usual orthonormal frame of
R 3
In order to express the equation defining the
minima of the energy one considers S2 as embedded in R 3. A point of S2 is then defined by three
coordinates nl, n2, n3 with the constraint n2 =1. The condition of local minimum of F (Eq. (2)) with the
constraint reads :
where A is the classical Laplacian operator acting on
a function in R3 :
One easily sees that the absolute minimum of the energy is obtained for n(r) = Const. The texture corresponds to uniformly oriented sample of nematic
molecules.
- S3 Blue Phase. - In a preceding paper, we defined the double twist condition of the blue phase
in terms of a connection V. We use, as in re-
ference [3], the isomorphism between the space
He 1 of quaternions with norm equal to 1 and S3. A point P of S3, considered with a radius R = 1, corresponds to the quatemion q :
with
The orthonormal frame field on S3 :
is a parallel frame field for the double twist connec-
tion.
The elastic free energy is expressed in terms of the
double twist connection [9] :
where gab is the Euclidean metric of R4. The director field n (with n2 = 1), expressed in the left invariant frame field given in equation (7), defines a map f : S3 -> S2. Since this frame field is parallel for the
double twist connection one gets :
and therefore the free energy F simply reads :
Extrema of F (f) are known [12] as harmonic maps f: 83 -+ S2. The condition of local minimum is :
where A is the
Laplace-Beltrami
operator acting onthe o-forms [10] n. A is expressed [9] with use of the
Levi-Civita connection V :
The frame field ea (Eq. (7)) is directed along
0
geodesics, oea (ea ) = 0 and the expression of A then
reduces to :
In order to find metastable textures of S3 blue
phase one must select, within each of the Z classes,
the harmonic maps. Our purpose is not to exhibit all harmonic representatives in every class of maps but
only to make explicit some of them i.e. those
corresponding to some specific Hopf invariant.
3. Director field and harmonic map.
3.1 RELATION BETWEEN VECTOR FIELD n (q ) AND S2. - Our purpose is now to construct vector fields
on S3 i.e. at each point q of S3 to find a unit vector n (q ) E Ts3 tangent space of S3. Let us clarify how to
relate a map f: S3 __+ S2 with the vector field
n(q)
The orthonormal frame field of equation (7), defin- ing a perfect S3 blue phase is left invariant [3] and generates an isomorphism between TS3, the Lie algebra (i, j, k) of S3 and R3 with its orthonormal frame (i, j, k). In the orthonormal frame of
TS3 the director field n ( q) reads :
or
This last expression specifies the components (V 1, V2, V 3) of the corresponding Lie algebra element.
The isomorphism
will allow us to deduce one vector field from one
map f.
3.2 CONSTRUCTION OF SOME HARMONIC MAPS. -
Metastable configurations are found by minimising
the free energy within each homotopy class. This condition selects harmonic maps f within each class.
A general procedure to select some of them results from the following construction. We proceed in two
steps as indicated in the diagram :
We first define a harmonic map h : S3 -+ S2 and then.
a harmonic map
The resulting map f : g o h is harmonic [11, 12].
- Hop f fibration h. - Let us first recall the polar
coordinates (rz, 6, a) of a point q of S3 widely used in
reference [9]
1864
Considering S3 embedded in R4, we can associate
to a point q E S3 an ordered pair of complex
numbers E C2, the complex 2-plane.
Writing equation (17) as :
one gets :
where
let us recall that k2 = - 1.
The Hopf fibration h can be deduced from a map
C2 - {0} --+ CP’ where CP’ is the one dimensional
complex projective space (i.e. space of all lines
through the origin in (2). h is then obtained by
identification of the unit sphere S2 with CP’ via the stereographic projection (see Fig. 1)
u , u2 u3 are the usual coordinates of P in
(R3. They can be expressed in terms of the polar
coordinates (rz, (J, a) of q
Fig. 1. - Stereographic projection. It links a point M of
the plane (R2 with coordinates (x, y) to a point P of S2 with coordinates (ul, u2, u3).
where
- Geometrical interpretation of the Hopf fib-
ration. - In the Hopf fibration given in equation (20), the set of points of S3 corresponding
to r2 and (0 - a ) constant, i. e. a great circle, does project on a single point of S2. It is easily seen by writing q (see Eq. (17)) in the following form :
with
When f3 and r2 are constant, q (J-L, r2, (3) can be
written as :
q(ix) = (cos g + k sin J-L ) qo where qo = Const.
This is the equation of a great circle. Its tangent
vector at point q (g ), p q ) X =
dq ,
is kq (IJ- ).d/i q( )
.
- Harmonic maps g on S2. - Harmonic maps g : S2 --> S2 can be defined from maps g’ :
where the integer p is the Brouwer degree of the mapping g’.
Coordinates on S2 are now :
which gives using the Hopf fibration of equation (20) :
4. Textures characterization.
The director field n is defined by the resulting
harmonic map f = g o h :
or
This director field generates a texture belonging to
one of the 7T3 (S2) = Z homotopy classes.
One way to specify the homotopy class of a
differential map is to look to related differential forms (cohomology). We refer the reader to re-
ference [13] for a mathematical presentation of the
de Rham cohomology. The main useful result is that
two homotopic maps induce forms which belong to
the same cohomology class. In a simple way one can say that we get information about differential maps f from one manifold X to another one Y
by looking at the pullback map f* on forms
fl (X) and n (Y) are the spaces of differential forms
on manifolds X and Y. Let us just recall that the de Rham cohomology deals with closed forms modulo exact forms. HP(Y) denotes an equivalence class of closed p forms. The Hopf invariant H(f ) is built with the aid of the map f* and depends only on the homotopy class of f.
- Hop f invariant. - The Hopf invariant is ob- tained in the following manner.
- Since each differential form 0 on Y = S2 of maximal degree, therefore closed, is a generator of
H2(S2), we take 6 = vol (S2 ) defined with the normalisation :
- The pull back form of 8, f * (8 ), is a closed two-
form a on S3. Since H2(S3) = 0, every 2-closed form
on S3 is an exact form and so :
Integration of equation (29) defines a 1-form w, not unique and a 3-form CO A dw on S3.
It is shown [13] that :
does not depend on the choice of w. This Hopf
invariant H(f) indexes the homotopy classes of f.
We can now proceed to the explicit computation
of H(f). In order to avoid difficulties due to the fact that two charts are necessary to define coordinates
on S2 and S3, one considers S2, S3 as embedded in R3, R4 and use the following coordinates :
where f : f’ o g’ o h’ and the pull back form f* = h’* o 9t * o f , *. Considering
this implies :
with
and
Integration of equation (31) leads to
and
with
1866
Using symmetry properties, H(f) can be expressed only in terms of the two first terms of equation (33).
or H (f ) = 4 p 3a where a is obtained with the change
u 1-t
as : u = - as:
t
This leads to the Hopf invariant
Let us point out that the procedure sketched in diagram 16 only exhibits maps in the classes indexed
by p2. This result is not intuitive. In that construction the Hopf fibration h has Hopf invariant H(h) = 1,
the harmonic map g has a degree D (g ) = p and the
resulting map f = g o h a Hopf invariant [14]
H (g o h) = P 2. One natural search for textures be-
longing to other classes would be to explore the
other side of the diagram :
Performing a harmonic map g of degree D (g ) = k
and then the Hopf fibration h the striking result is
that the resulting map 1 of Hopf invariant H(f ) = k
is not harmonic. It is unknown whether classes with
Hopf invariant =F k2 have harmonic representatives [11, 12].
- Geometrical interpretation of the Hopf in-
variant. - Let us just consider the points of S3 giving an image :
on S2.
The general expression for Vi is :
Comparison with the map f defined by equation (28) gives the inverse image of a point P of S2 (defined by cxi 1 and u 1 = Const.). The condition a = Const.
implies T = Const., i.e. (Eq. (27)) r2 = Const. This
defines a torus on S3.
For fixed tk 1 one obtains :
or
Equation (38) gives p values of (0 - a) and then generates p non intersecting great circles on the
torus corresponding to r2 = Const. In the same way the inverse image of an another point q V 2 of S2 is a new set of p circles. Let us recall that any great circle on S3 links any other (non intersecting) great circle (Appendix). This implies that any each of the p great circles of the first family links each circle of the second family. It defines a linking number
p2. This procedure is general ; for any map this
linking number is the Hopf invariant.
, -
Energy of harmonic textures. - A straightfor-
ward calculation shows that the director fields con-
structed in section 3 are indeed harmonic. The
expression of the Laplace operator in polar coordi-
nates (r2, a, 0) is :
I
The condition of local minimum Ani(P) =
A (P) ni(p) is verified for :
The energy is calculated with use of polar coor-
dinates :
where
and
One obtains :
and after integration :
The fields we have built correspond to metastable
structures. Up to now we have not succeeded to
show that the energy given in equation (44) is an
absolute minimum. From a physical point of view
this question is important but its mathematical
resolution is difficult. Nevertheless metastable tex- tures are worth describing.
- Director field lines. - Before performing the general analysis we first look for some specific
director fields. Let us consider the case p = 0 which
corresponds to a constant vector field :
It describes a typical field belonging to the homotopy class p = 0. Every field associated with a
constant element of the Lie algebra is in the same homotopy class (for example the field no = qk we
shall use in the following). Another class, with
p = 1, leads to an easy illustration. One director field in this class is :
Choosing cpo = m /2 allows us to recover a particular
field already known as :
These particular director fields no and nl, as shown in figure 2, lie on the surface of the tori defined by
r2 = Const. no is tangent to great circles correspond- ing to 0 + a = Const. 81 is tangent to great circles corresponding to (J - a = Const. One must be aw- are of the following aspect. From the geometrical point of view given just before the two director fields no, 81 look similar. But the construction given by equations (14) and (15) corresponds to left invariant
fields such as no linked to a particular sign of the
twist. With that choice of twist, no describes a perfect phase of energy equal to zero. n, is related to a twist of opposite sign with a non zero energy given in equation (44).
Fig. 2. - Special director fields belonging to the two homotopy classes p = 0 and p = 1. no (p = 0) and 81 (p = 1 ) lie on the torus defined by r2 = Const., rep-
resented by the rectangle where opposite sides are iden-
tified. One moves on the torus by changing 0 and a from 0
to 2 1T (circles on the torus). no and 81 are tangent to the
two great circles Co, C1 defined respectively by
a + 0 = Const. and a - (J = Const.
In the general case, director field lines are ex-
pressed in a convenient manner with use of the frame field
In this frame the director field reads :
with
and director field lines satisfy :
where dr stands for dr2. To simplify the comparison
with the case p = 1, we also choose ’Po = ?r/2 in equation (47).
The analytical expression of director field lines
implies the complete resolution of system (48) which
is not straightforward in the general case. But partial integration of equation (48) can be performed for
any p and leads to :
where B =Const. > 0.
Each value of B fixes some field lines the extension of which is given by the’ condition :
For each value of B -- 2P the resolution of this
inequality implies that :
where the precise value of ro depends on p. Director field lines are then confined between the tori fixed
by ro and
J 1 -
ro. They do not remain in generalon the surface of the tori r2 = Const., except in two
cases :
- B=0.
Director field lines only exist on the spherical
