Umbilics in orientational order
J. Binysh1
1 Centre
for Complexity Science
University of Warwick
March 7, 2016
Liquid Crystals
I
1.1 Elastic distortions
Described by a director
field, n(r ).
I
|n| = 1, n ∼ −n, i.e n ∈ RP 2 .
Figure 1.1: An exemplary Schlieren texture in a nematic liquid crystal. Points with
brushes correspond to disclinations with topological strength s = 1/2 and s = 1,
Reproduced with the kind permission of Ingo Dierking.
The points themselves correspond to singularities, or defects, in the orientation
Liquid Crystals
966
NATURE MATERIALS DOI: 10.1038/NMAT2592
a
b
P
A
T3-3
T3-2
d
T3-1
10 µm
T3-2
f
Figure 2 | Predetermined optical generation and switching of the toron structures. a, Po
two T3-2s of opposite winding (intermediate size) and T3-3 (the largest structure) genera
by optical generation of four T3-2s per letter at the letters’ vertices and T3-1 elsewhere wi
FIG. 5.
M.B.B.A. additionné d’un faible pourcentage de benzoate de cholestérol, examiné entre
and
analyser (A) are shown by the white bars. b, Polarizing microscopy image showing tha
frottés parallèles.
Utilisation
d’un polariseur perpendiculaire à la direction de frottement sauf pour les figures m et où le
polariseur parallèle frottement et l’observation est faite
contraste de phase. Pour le détail des figures, voir le texte. Les échelles sont pratiquement les mêmes
la figure 1.
generated
at an arbitrary location in the sample and then moved to the desired position (s
que pour
Chaque segment représente 20 y.
shifting the generating infrared laser beam. c, After optically moving the T3-1 to the image
-
verres
n,
en
est
au
Energetics of Liquid Crystals
I
Free energy naturally described by the geometry of the
system - gradients of the director.
Z
K2
K3
· n)2 +
(n · ∇ × n + q0 )2 +
((n · ∇)n)2
2
2
{z } |
{z
} |
{z
}
K1
(∇
F =
1.1 Elastic distortions
2
|
Twist
Splay
Bend
Figure 1.2: Schematic illustration of the elastic distortions in a nematic liquid crystal.
following expressions being non-zero:
3
Defects
Downloaded from rsta.royalsocietypublishing.org on March 19, 2013
(a)
(b)
5 µm
(f)
5 µm
Figure 2. A silica microsphere, treated so as to align the LC molecules perpendicular to its surface, appears in the form of either a
dipole (a) or a quadrupole (d). (a,d) Photographs taken without any polarizers. The small dot in (a) is the hedgehog topological
defect. The ring, visible in (d), is the Saturn ring. It is a −1/2 disclination ring, encircling the particle at the equator. The images
I inPeople
have
thought
about
textures
in liquid
crystals
etc.
(b) and (e) are taken
using crossed
polarizers and
the red (lambda)
plate. (c,d) A schematic
of the director-field
distribution.
(Online version in colour.)
by thinking about defects
nematic LC for normal surface-anchoring conditions. The first one has the symmetry of a dipole
......................................................
5 µm
(e)
4
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120266
5 µm
(d)
(c)
Classification of defects
Homotopy Theory
Alexander et al.: Colloquium: Disclination loops, point . . .
I
Classifying continuous maps from real space into order
parameter space, using homotopy theory.
π1 (RP 2 ) ' Z2
π2 (RP 2 ) ' Z
FIG. 3. The GSM for the three-dimensional nematic is the real
projective plane RP2 . Below each hemisphere which represents
RP2 we draw the path in the sample and the corresponding director
at each point, with the arrows on the two curves denoting corresponding points and orientations. Any measurement around a closed
503
FIG. 1 (color). (a) Selected 3PEF-PM images from an image stack—the images are 16 m wide and approximately 4 m apart in
the z direction. (b) The toron texture from Ref. [12]. The axis of symmetry of this structure is perpendicular to the substrate. (c) Top
view, (d) side view: the ‘‘Pontryagin-Thom’’ surface constructed from this image stack. Despite the considerable noise, the robustness
of this method allows us to recognize the nontrivial topology of the texture as the winding bands of color meeting at the two hedgehog
defects on top and bottom.
Non trivial textures with no defects
The Hopf fibration
NATURE
10.1038/NMAT2592
ARTICLES
1 ðRP2 Þ, while a point boundary,
i.e.,MATERIALS
a hole in theDOI:
surface,
through the color wheel a second time if the director
rotates
carries a Z charge associated with 2 ðRP2 Þ. To illustrate,
by 2, as it does, forb instance, in z^ cfor the standard radial
a
l=9
3
P
the surface z^ for a toron is shown in Figs. 1(c) and 1(d).
hedgehog
shown in Fig. 3. Inl =fact,
since all point
defects in
A
T3-3 the director
It is the set of all points in the material where
a uniaxial nematic can be oriented [16], the usual n^ ! n^
T3-2
20 µm
is perpendicular to z^ and forms a surface that connects
symmetry
does not come
into play, and the director always
the two point defects at the ‘‘top’’ and ‘‘bottom’’ of
rotates through the color
wheel an even
number of times,
e
d
8
l=5
the toron.
that is, by a multiple of 2.l =As
a result, a point
defect of
The surfaces, however, do not carry enough information
charge p 2 2 ðRP2 Þ will have a winding of 2p, or will
T3-1
10 µm
to determine the point charges. In order to capture this
cover the color wheel 2p times, providing a unique idenT3-2 of informaf
g Thus,l =looking
information, we must add an additional piece
tification of point defects
in l nematics.
at the
9
=3
tion to the surface, namely, the direction of the director in
toron in Figs. 1(c) and 1(d) again, the surface z^ is colored
p^ . We represent this pictorially through a color wheel,
as described, and about each point defect there is a twofold
winding of the full color wheel, identifying them as carryranging from red to violet—through orange, yellow, green,
ingofunit
charge.
blue, and indigo—as the director
rotates byoptical
. generation
We pass
Figure 2 | Predetermined
and switching
the toron
structures. a, Polarizing optical microscopy texture showing T3-1 (the smallest),
two T3-2s of opposite winding (intermediate size) and T3-3 (the largest structure) generated next to each other. The inset shows the letters ‘CU’ obtained
by optical generation of four T3-2s per letter at the letters’ vertices and T3-1 elsewhere within the characters. The orientations of the crossed polarizer (P)
and analyser (A) are shown by the white bars. b, Polarizing microscopy image showing that the two T3-2s of opposite spiralling and the T3-1 can be
generated at an arbitrary location in the sample and then moved to the desired position (such as the one shown in c) using optical manipulation by laterally
shifting the generating infrared laser beam. c, After optically moving the T3-1 to the image centre, the new T3-3 structure is generated in the top right
corner of the image. d–g, This T3-3 structure is shown transforming into a T3-2 structure (d,e), then to T3-1 (f) and again to the T3-3 configuration (g) by
using Laguerre–Gaussian beams of appropriate topological charge values l marked on the images. Note that the T3-3 structures in c,d,g have different
diameters of the disclination rings at the top and bottom surfaces and all T3-2s and T3-3s have lateral dimensions 1.1–1.5 times larger than T3-1s.
a
c
ẑ
ŷ
x̂
xy
10 µm
d
d
FIG. 2 (color). (a) A simulation of a toron in which the point hedgehogs are replaced with disclination
loops. (b) Flow lines of the
5 µm
famous Hopf fibration. (c) The preimage surface of the Hopf fibration, which one can get
from (a) by bringing together the two
d
xz
5 µm
ẑ
disclination loops through the center of the spool. (d) An experimental image of the same texture.
I
b
xz
237801-2
Discontinuities don’t tell us
everything. Here’s a
topologically non trivial texture containing no defects.
e
I
π3
(RP 2 )
'Z
xz
Non trivial textures with no defects
Lambda lines
UKH AND O. D. LAVRENTOVICH
PHYSICAL REVIEW E 66, 051703 共2002兲
CPM texturesIof Not
verticalsingularities
x-z cross sections in
of the
Grandjean-Cano
with right-handed CLC,
p⫽5 ␮ m, strong planar
directorwedge
- degeneracies
in the
twist disclination separating 0 ␲ and 1 ␲ Grandjean zones; 共b兲 b⫽p/2 dislocation with a core split into a ␶ ⫺1/2␭ ⫹1/2
Umbilic
ir, separating 2 ␲ derivatives.
and 3 ␲ Grandjean zones;
共c兲 the lines.
same, between 13␲ and 14␲ zones; 共d兲 b⫽p dislocation with a core split
⫹1/2
disclination pair, 22␲ and 24␲ zones, the slice obtained along the bb line in Fig. 3共b兲. Polarizer P is parallel to the y axis
irection is along the x axis. Bright regions correspond to n兩兩 P, darker regions correspond to n⬜P or bounding glass plates.
Using the differential structure
I
The texture is more than continuous - it’s differentiable.
I
The derivatives have obvious structural importance
Z
K1
K2
K3
F =
(∇ · n)2 +
(n · ∇ × n + q0 )2 +
((n · ∇)n)2
2
2
2
I
Rather than looking at n(r ) alone, also look at ∇n(r ).
These gradients contain structural, topological information.
The structure of ∇n
I
At each point n(r ) defines a line and a plane - it splits the
tangent bundle.
T R3 ' Ln ⊕ ξ
I
∇n eats an element of T R3 , and returns a direction in the
orthogonal plane ξ
∇n ∈ Γ(T ∗ R3 ⊗ ξ) ' Γ((L∗n ⊗ ξ) ⊕ (ξ ∗ ⊗ ξ))
∇n in a local frame
I
I
We see this explicitly by expressing ∇n in a nice local
frame. Let {d1 , d2 } ∈ ξ. Then {n, d1 , d2 } form a local basis.
Two parts - bend (derivatives in the direction of n), and
everything else.
n
d2
d1
∇n ∈ Γ(T ∗ R3 ⊗ ξ) ' Γ((L∗n ⊗ ξ) ⊕ (ξ ∗ ⊗ ξ))


0 (n · ∇)n · d1 (n · ∇)n · d2
0 (d1 · ∇)n · d1 (d1 · ∇)n · d2 
0 (d2 · ∇)n · d1 (d2 · ∇)n · d2
Umbilic Lines
I
Lets focus on the part that lives in (ξ ∗ ⊗ ξ). Call it ∇⊥ n. Its
analogous to the shape operator.
I
The action of SO(2) on (ξ ∗ ⊗ ξ) gives a natural way to
further decompose ∇⊥ n.
1
1
1 0
0 −1
∆1 ∆2
+ n·∇ × n
+
∇⊥ n = ∇ · n
∆2 −∆1
0 1
1 0
2
2
(ξ ∗ ⊗ ξ) = I ⊕ J ⊕ E
I
The places where ∆ ∈ Γ(E) vanishes are called Umbilic
Lines. These turn out to contain structural information
about the vector bundle ξ.
I
∆ really does vanish along lines. 2 constraints in R3 .
zero, so you can scale it to have unit length. Then
Zeroes
andfield
Topology
r of
the vector
around the zero is characterised
I Let’s
motivate number,
why looking
at themap.
zeroes
of ∆ is interesting.
the degree,
or winding
of that
Again,
I
An
example
of
the
connections
between
know whether the vector field is winding positively
I zeroes of a section of a vector bundle
the vector bundle
(both base and fibre) around the
I structure of a vector bundle
e indices of two
or
more
zeros we need to extend this
I topology of a space
en, computing
of theorem.
the indices of the zeros of
Is the the
Hopfsum
index
oyable.
os on the sphere
thout any. Here
pole with some
ns trying to hold
ngs to straighten
s ‘travel along a
d what we mean
allel transport’2 .
arkably enlightbe non-zero ev-
Everywhere on this little circle the vector is nonzero,
so you
canTheorem
scale it to have unit length. Then
Hopf
Index
r of
the
vector
field
around the zero is characterised
aka "the hairy ball theorem"
the degree, or winding number, of that map. Again,
X
know whether the vector field
is xwinding
positively
index
(v ) = χ(M)
i
the vector bundle (both base
and fibre) around the
i
e indices of two or more zeros we need to extend this 2
I We cannot construct a nonzero section of T (S ).
en, computing
the sum of the indices of the zeros of
I Why we can’t is related to χ(S2 ), a purely topological thing.
oyable.
os on the sphere
thout any. Here
pole with some
ns trying to hold
ngs to straighten
s ‘travel along a
d what we mean
allel transport’2 .
arkably enlightbe non-zero ev-
u go around a little circle about the zero.
on
this Index
little circle
the vector is nonHopf
Theorem
u can scale it to have unit length. Then
Abovethe
wezero
hadisacharacterised
surface, S2 , and its tangent bundle T S2 .
tor field I
around
or winding
number, of constructions
that map. Again,
I Analogous
exist for a general vector bundle
her the vector
is winding
positively
overfield
a general
space.
In particular, for the bundle ξ (and
bundle (both
and fibre)byaround
the field.
(ξ ∗base
⊗ ξ))defined
the director
two or more
zeros
westructural
need to extend
this about the bundle ξ by
I We
learn
information
ing the sumstudying
of the indices
of
the
zeros
the zeroes of ∆. of
here
Here
some
hold
hten
ng a
mean
ort’2 .
ighto ev-
What I’m doing
I
I
∇⊥ n ∈ Γ(ξ ∗ ⊗ ξ) has been partially analysed.
I’m beginning work on similar constructions for (L∗n ⊗ ξ),
and seeing how they relate to one another.
∇n ∈ Γ(T ∗ R3 ⊗ ξ) ' Γ((L∗n ⊗ ξ) ⊕ (ξ ∗ ⊗ ξ))


0 (n · ∇)n · d1 (n · ∇)n · d2
0 (d1 · ∇)n · d1 (d1 · ∇)n · d2 
0 (d2 · ∇)n · d1 (d2 · ∇)n · d2