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Disclination in the S3 blue phase
B. Pansu, E. Dubois-Violette, R. Dandoloff
To cite this version:
B. Pansu, E. Dubois-Violette, R. Dandoloff. Disclination in the S3 blue phase. Journal de Physique, 1987, 48 (2), pp.305-317. �10.1051/jphys:01987004802030500�. �jpa-00210444�
Disclination in the S3 blue phase
B. Pansu, E. Dubois-Violette and R. Dandoloff (*)
Laboratoire de Physique des Solides, Bât. 510, Université Paris-Sud, Centre d’Orsay, 91405 Orsay, France (Reçu le 23 mai 1986, revise le 9 septembre, accepté le 7 octobre 1986)
Résumé. 2014 Nous donnons une description détaillée de la phase bleue sur S3 en termes de feuilletage de Hopf. Celle-
ci peut être vue comme un nématique dans l’algèbre de Lie correspondante. Utilisant cette analogie, nous
construisons une disinclinaison dans la phase bleue sur S3 par un processus de Volterra semblable à celui du
nématique. Nous donnons les caractéristiques essentielles du champ disincliné ainsi obtenu et calculons l’énergie élastique associée.
Abstract. 2014 The perfect S3 blue phase is described in details in terms of Hopf foliation. It can be seen as a nematic
phase in the corresponding Lie algebra. From this point of view we introduce a disclination in the S3 blue phase by a
Volterra process similar to the nematic case. We give the main features of the resulting director field and compute the elastic energy.
Classification
Physics Abstracts
61.30 - 61.70G
1. Introduction.
Blue phases of chiral nematic liquid crystals are frus-
trated systems in R3. Some of these phases present a crystalline cubic structure. Meiboom, Sammon and
Brinkman [1] proposed a model interpreting these phases. The condition of double twist [2] cannot be
realized everywhere in R3. It can be satisfied locally on cylinders of finite size directed along the three main
directions of a cube. Inside the cylinders the molecules
twist radially. The director orientations fit together at
the points where two different cylinders touch each
other. This imposes an array of disclinations of S = - 1/2 order in some diagonal directions.
Another description of these phases has been given by Sethna [2] who has shown that the double twist condition can be realized everywhere in a 3 dimensional
sphere S3.
The link between the two models was done geometri- cally by Sadoc [3]. In the model of Meiboom et al., the director field close to the axis of the cylinders is similar
to the director field of the perfect S3 blue phase. This
can be seen by using a particular foliation of S3 that we
shall describe in detail in section 2. This correspondence
is only local, a global one would imply to relieve the
curvature of S3.
The purpose of this paper is not to flatten all the space S3 but to introduce a single defect in a perfect
double twisted structure on S3. A similar approach has
been developed by Sadoc and Charvolin [5] for bilayers
of amphiphilic molecules. For the blue phase we realize
this process in an easy way by using the analogy developed in reference [4] : the S3 blue phase appears as
a nematic one in the Lie algebra OfS3 . This main point is
revealed by the diffeomorphism between the unitary quaternion group H1 and S3. This diffeomorphism is
also very convenient to build a disclination line on
S3. To emphasize this similarity we first recall in
section 3 the Volterra process generating a singular line
inside a perfect nematic. Then we use a trick (we work
in the Lie algebra of H1) to introduce in a similar way a
singular loop in the S3 blue phase. Whereas the Hopf
foliation nicely pictures the perfect state, there is no such global description for the distorted one. We give
in section 3 the analytical expression of the disclinated held and then focus our attention in section 4 to some
relevant features. In that spirit we describe in detail the director held on some particular surfaces and also very
near to the disclination line. Since locally S3 is isomorp-
hic to R 3 we thus obtain a configuration, close to the line, topologically equivalent to the R 3 one [1]. The
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802030500
double twist connection we constructed in a preceding
paper [4] allows a rapid computation of the elastic free energy (Sect. 5).
2. Description of the perfect S3 blue phase.
In a previous paper [4] we described a frame field satisfying the double twist condition on S3. In what
follows we shall use the unitary quaternion group notation (Appendix B). In order to describe the director field lines we consider one particular tangent
vector field satisfying the above condition. Let us just
consider the vector field associated to the Lie algebra
element V = i. Then at any point q of the sphere S3,
the value of the vector field is X = qi. The director
field lines are tangent to this particular vector field to
which we shall always refer in the following as the perfect ordered phase. A first geometrical approach of
this field was given by Sadoc [3] in terms of Clifford lines (Fig. 1) (in the following we shall use the term Hopf foliation : the leaves of this foliation are the fibers of the Hopf fibration [14]). The sphere S3 is covered by
a family of tori nested in one another. Each torus may be generated by a family of great circles. Then, in that foliation, coordinates of a point q of S3 are given by the
Fig. 1. - The S3 blue phase director field lines belong to a family of tori. We reproduce a picture first given by Sadoc [3].
knowledge of one torus, one great circle on this torus, and one point on this great circle. We shall now describe some characteristic lines of this foliation and surfaces which will appear useful later on for the defect construction.
2.1 GREAT CIRCLE. - A great circle C is defined by a
curve :
where Xo is a unit imaginary quaternion. At each point
q ( t ), the vector tangent to this circle is :
In what follows we shall denote such a circle by
C = (qo, Xo) referring to one point qo of C and to the
Lie algebra element Xo associated with the tangent
vectors. The director field lines of the S3 blue phase
defined above correspond to circles with X6 = i.
Let us point out that, associated to any circle
C, = (q,, Xo) , there exists another circle
C2 = ( q,, xo ) perpendicular to Cl. For example :
and
A great circle may also be defined by the intersection of a 2D plane of o¡4 containing the origin with the S3 sphere. Another type of circle (of smaller radius) would correspond to the intersection of S3 with a 2-plane hot containing the origin.
Stereographic projection of a circle (Appendix A) is
either a circle or a straight line (for a circle passing through the pole). The three axes of the projection correspond to the three circles (1, i), (1, j), (1, k).
2.2 S2 SPHERE ON S3. - It corresponds to the intersec-
tion of a 3-plane of R4 with the sphere S3 : it is a 2 dimensional surface defined by the following two equations :
If one imposes a2 + b2 + c2 + d2 = 1 and combines
these equations, one obtains :
This proves that all the points (XO, xi , x2, X3) of the
sphere S2 defined by equations (4) are at the Euclidean
distance d1 = /2 ( 1-- f ) of a point A of S3 with
coordinates (a, b, c, d) and at the distance
d2=N/2(1+f) of the point B = - A of S3. If the
3D plane of R4, which intersects S3 to give the sphere S2, contains the origin 0 of 1R4, this sphere is a great
sphere : if it contains q E S3, it also contains - q. A great sphere is at the same distance d1 = d2 = J2 of
point A and of the antipodal point - A: The stereo- graphic projection of such a sphere S2 is either a sphere S2 of R3 or a 2D plane.
2.3 TORUS. - It corresponds to a set of points of S3 at a distance d1 of a great circle Cl and at a distance d2 of another great circle C2 perpendicular to Cl. We
shall call one of these circles the axis of the torus. A torus in S3, is a 2D surface. Taking for the circles Cl =
( 1, i ) and C2 = ( k, i ) one describes the torus as :
with tl > 0, t2 > o, tl + t2 =1. The distances to the axes
of the torus are d1 (2) =1- t1 ( 2 ) . The spherical
torus corresponds to d 1 d2, ti = t2 = .J2’/2
The torus q ( À , IL) is the direct product of two circles, a first one (Fig. 2) :
q 1 ( À) = t 1 ( cos À + i sin À) in the plane ( 1, i )
and a second one
q2 (u ) = t2 (j COS IL + k sin IL) in the plane ( j, k ) .
We shall now give another parametrization which enlightens some other interesting properties.
Equation (5) may be written as :
where
and
One recognizes in expression (7) the equation of a great circle (qA’ i’) where the point qA = tl - t2 k is the
intersection of the torus with the circle (1, k). One
moves along circle C2 by changing a. Through each point of C2 passes another great circle qq, i
(Eq. (6)) see figure 3. In the S3 blue phase this great circle corresponds to a director field line. One generates
the torus by moving the circle (qA’ i) along the circle
C2 (Fig. 3). This corresponds to the--geometrical de-
Fig. 2. - The stereographic projection of a S3 torus is a S2
torus. The x axis corresponds to the projection of the circle
(1, i). This torus is the product of two circles of radius tl and t2, A and u describe the phases on the circles.
Fig. 3. - Stereo graphic projection of a S3 torus. Through
each point qc2 of the circle C2 on the T2 torus passes another great circle of tangent vector qc2 i. The torus is generated by moving the circle ( qA, i ) along the circle C’2.
scription of the S3 blue phase of figure 1 (represented in
a stereographic projection where a S3 torus is projected
on a R3 torus). Varying tl from 0 to 1, one creates a
family of tori successively nested in one another. The two extreme tori are the two great circles
and
This foliation allows us to visualize the director field lines as pictured in figure 1.
3. Introduction of a defect. 1/2.
The basic idea is to use once more the similarity emphasized in reference [4] between the nematic in
R3 and the blue phase in S3. The fundamental point is
that in both cases the ordered phase is associated with a
constant element of the Lie algebra. We first recall how
a disclination line can be introduced in a nematic
sample and then extend the method to the S3 blue
phase with use of the analogy stated above.
3.1 DEFECTS = - 1/2 IN A R 3 NEMATIC. - A simple geometrical process to introduce this disclination line £ is the following. Consider a perfect ordered nematic
configuration defined by n ( TO) = Const. Since we
describe the simplest defect line S = - 1/2 (straight line
in the z direction) we also introduce the simplest
surface (a half plane yOz, y > 0) along which we cut the sample. We separate the two lips of the cut surface and
introduce in the void space some extra (half space) perfect nematic matter. By relaxation this process generates the singular line S = - 1/2 [6].
In this process one half space is transformed into two-thirds of space. We choose a mathematical descrip-
tion that can also be extended to the S3 blue phase. The
new configuration is obtained from the initial one by rotating both each point of the space and the associated director. The angle of this rotation only depends on the polar angle 0 around the disclination line. The origin of
the 0 angle is taken along the Oy axis (Fig. 4). The
transformation r on the space point is :
This corresponds to a rotation % of angle
around the z axis.
Performing this rotation on the initial state n = i, we obtain the disclinated director field :
where (i, j, k) is the Cartesian frame.
We now give another construction of the disclinated director field, which will be useful for comparison with
Fig. 4. - Schematic representation of the Volterra process around the line Oz.
the S3 blue phase. Then in such polar frame
( e r’ e 8’ e z ) .
In these coordinates the introduction of the disclination line corresponds to the change 0 - 3 0/2.
3.2 DEFECT S = - 1/2 IN A S3 BLUE PHASE. - In the nematic we introduced easily a straight disclination line with use of cylindrical coordinates. By analogy we
define coordinates ( r 2’ (J, a) in S3 revealing the
angular symmetry around the line. Let us consider the circle (1, k) as the disclination line, then these coordi- nates are :
where we use the notations (see Appendix B) :
for
The choice for the 0 origin (Fig. 5) will appear clearly
when we shall introduce the cut surface. The distance
d2 from a point q to the great circle (1, k) is d2 =
J 2 ( 1 - r 2) . r 2 plays the same role as r in .the
cylindrical coordinates of R3 introduced in the nematic
case. The surfaces r2 = Const. are tori (same role as cylinders in R3). Surfaces defined by a = Const. are
Fig. 5. - The origin of the 0 angles is taken along the axis Oy corresponding to the stereographic projection of the great circle (1, j). 0 is taken in {0, 2 ir).
great spheres in S3 perpendicular to the disclination line. Their stereo graphic projections are either 2 sphere
or plane (for a = 0 or 1T). a e {0, 2 ir) plays the same
role as z in R3, and the spheres the same one as planes
in R3. It is now clear that 0 is a polar angle around the
line (as in R3). It is also the polar angle around the z
axis in the stereo graphic projection (the z axis coffes- ponds to the projection of the disclination line).
Surfaces 0 = Const. are half great spheres in S3 con- taining the disclination line the projection of which are planes (0 = Const.).
We introduce the cut surface by considering the S3 sphere embedded in R 4. In nematics the cut surface
was a 2D plane (passing through the line). By analogy
we take, as cut surface (containing the great circle), the
intersection of S3 with a 3D plane of R4 (containing the origin) i.e. a great sphere [7]. The cut surface, where
we shall introduce a half space to create the disclination, corresponds to 0 = 1T (the origin of the 0 angles has
been taken along the axis Oy) see figure 4. The
intersection of the cut surface with each torus of the foliation (1, i) is a half small circle whose stereographic projection is a circle, as shown in figure 6.
We now want to describe the operation, equivalent
to the rotation around the disclination line introduced in the nematic. We choose the rotation R of angle 0 in R4 which leaves the plane containing the disclination line (great circle (1, k)) invariant. This reads :
with
This induces the following change in polar coordinates
Fig. 6. - Stereographic projection of the cut surface defined
by 0 = ?r. The intersection of this cut surface and one torus of the foliation (1, i) is a half small circle projected on C.
- As shown in the appendix, an interesting property of the quaternion group H1 (isomorphic to S3) is that
tangent vectors X do transform in the same way as the
points of the manifold.
Then
- As shown in a preceding paper [4], the tangent vector X at a point q may be expressed in terms of Lie algebra element V as :
Using equations (11) (13) we show that 3t also induces the same rotation on V as on a quaternion :
The isomorphism between the Lie algebra of the quaternion group (imaginary quaternions) and R3 al-
lows us to associate the vector T = V x i + Vy j + V z k
of R3 to the element V. Then to the element V’
deduced from V by :
where
we associate the vector T’ = V x i + V y j + V i k of 1R3.
T’ is obtained from T by a rotation of angle 0 around
the axis u ( ux, uy, uz) -
Precisely the quaternion q1 (Eq. (12)) is related to a
rotation in R3 of angle 0 around the k axis. Applying equation (16) to the S3 blue phase defined by V = i, we obtain :
We can now proceed to a specific construction of a
disclination of order - 1/2. Using once more the
analogy with the nematic, we define a similar Volterra process. The introduction of a half space of perfect S3 blue phase between the two lips of the cut surface
creates a defect with three fold symmetry. The new field at a point q ( r, 0, a) is obtained from the
previous perfect one at point qo (’0’ (Jo, a o ) :
Point q is obtained from point qo by a transformation of type (11) where 0, defined in equation (12), is 0 = o - 0 = - 00- 0. The perfect initial S3 blue
= N - eo =-2013==-- The perfect initial S3 blue
phase was associated with a constant element of the Lie
algebra V = i defining the vector field qV. The new
vector field, after introduction of the defect, is (Eq.
(18)) now qi’ where :
As claimed in reference [4] the perfect blue phase can
be viewed in terms of a perfect nematic configuration in
the S3 Lie algebra. This analogy also stands for a
distorted state. The disclinated field . qi’ is associated to the Lie algebra element i’ (Eq. (20)) identical to the
disclinated field n’ (Eq. (8)) of the nematic if we
identify the orthonormal frame (i, j, k) of R3 with the
basis, i, j, k of the S3 Lie algebra.
4. Description of the distorted S3 Blue phase.
In section 2, we have given a geometrical description of
the perfect S3 Blue phase in terms of « Hopf foliation ».
It is not possible to describe the distorted state in such a simple way. With the analysis given in the preceding
section the director field can be computed everywhere.
Now we shall only describe relevant features of this field.
4.1 PARTICULAR FIELD LINES. - Some special lines
reveal the symmetry properties of the director field.
The disclination line we introduced cut some great circles of the Hopf foliation into two half circles (see Fig. 7). The two points A(a) and B(a), intersections of the line with these circles, are defined by coordinates
+ a and 2 7r - a. The two half circles are the intersec- tions of S3 with two half spheres defined at each point a
of the line by :
In the Volterra process, each half sphere 0 (a) =
Const. is transformed into another half sphere ·
Therefore each pair of half circles gives rise to three
new half great circles (Fig. 7) :
Equation (23) describes how the director twists along
the disclination line. In figure 8 comparison is shown
between the two situations : before and after introduc- tion of the defect.
Fig. 7. - The introduction of a disclination line transforms the two half circles (a) into the three half circles (b). The tangent vectors are indicated with dotted lines case a) and
bold lines case b).
4.2 DISTORTED TORI AND RELATED FIELD LINES. - In the perfect state, director field lines lie on the surface of tori nested in one another. The particular
process we use to introduce the disclination transforms the above family of tori into a new family of « tori »
with three lobes (Fig. 9) nested in one another. These
new « tori » are deformed : we indicate, in the insert of figure 8, the section in the plane x0y of tori of different sizes. We represent on each lobe the director field lines which are not distorted in the process (half circles of
