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Crystals and liquid crystals confined to curved geometries

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This review introduces the elasticity theory of two-dimensional crystals and nematic liquid crystals on curved surfaces, the energetics of topological defects (disclinations, dislocations and pleats) in these ordered phases, and the interaction of defects with the underlying curvature. This chapter concludes with two cases of three-dimensional nematic phases confined to spaces with curved boundaries, namely a torus and a spherical shell.
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Crystals and liquid crystals confined to curved geometries
Vinzenz Koning and Vincenzo Vitelli
Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The Netherlands
Abstract
This review introduces the elasticity theory of two-dimensional crystals and nematic liquid crystals
on curved surfaces, the energetics of topological defects (disclinations, dislocations and pleats) in
these ordered phases, and the interaction of defects with the underlying curvature. This chapter
concludes with two cases of three-dimensional nematic phases confined to spaces with curved
boundaries, namely a torus and a spherical shell.
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arXiv:1401.4957v1 [cond-mat.soft] 20 Jan 2014
CONTENTS
I. Introduction 3
II. Crystalline solids and liquid crystals 5
III. Differential geometry of surfaces 6
A. Preliminaries 6
B. Curvature 8
C. Monge gauge 10
IV. Elasticity on curved surfaces and in confined geometries 10
A. Elasticity of a two-dimensional nematic liquid crystal 10
B. Elasticity of a two-dimensional solid 12
C. Elasticity of a three-dimensional nematic liquid crystal 14
V. Topological defects 14
A. Disclinations in a nematic 15
B. Disclinations in a crystal 15
C. Dislocations 17
VI. Interaction between curvature and defects 17
A. Coupling in liquid crystals 17
B. Coupling in crystals 19
C. Screening by dislocations and pleats 20
D. Geometrical potentials and forces 20
VII. Nematics in spherical geometries 22
A. Nematic order on the sphere 22
B. Beyond two dimensions: spherical nematic shells 23
VIII. Toroidal nematics 25
IX. Concluding remarks 26
References 27
2
I. INTRODUCTION
Whether it concerns biological matter such as membranes, DNA and viruses, or syn-
thesised anisotropic colloidal particles, the deformations inherent to soft matter almost in-
evitably call for a geometric description. Therefore, the use of geometry has always been
essential in our understanding of the physics of soft matter. However, only recently geometry
has turned into an instrument for the design and engineering of micron scaled materials. Key
concepts are geometrical frustration and the topological defects that are often a consequence
of this frustration [1–3].
Geometrical frustration refers to the impossibility of local order to propagate throughout
a chosen space. This impossibility is of geometric nature and could for instance be due
to the topology of the space. Probably your first and most familiar encounter with this
phenomenon was while playing (association) football. The mathematically inclined amongst
you may have wandered off during the game and wondered: “Why does the ball contain
hexagonal and pentagonal panels?” The ball cannot merely contain hexagonal panels: a
perfect tiling of hexagons (an example of local order) cannot be achieved on the spherical
surface (the space considered). There exists a constraint on the number of faces, F, edges,
E, and vertices, V. The constraint is named after Euler and reads [4]
FE+V=χ, (1)
where χis the Euler characteristic. The Euler characteristic is a quantity insensitive to
continuous deformations of the surface of the ball such as twisting and bending. We call
such quantities topological. Only if one would perform violent operations such as cutting a
hole in the sphere and glueing a handle to the hole a surface of differently topology can be
created [4, 5]. For a surface with one handle χ= 0, just as for a torus or a coffee mug. The
Euler characteristic χequals 2 for the spherical surface of the ball. Thus, Euler’s polyhedral
formula (eq. (1)) ensures the need of 12 pentagonal patches besides the hexagonal ones, no
matter how well inflated the ball is. To see this, write the number of faces Fas the sum of
the number of hexagons, H, and pentagons, P, i.e. F=H+P. One edge is shared by two
faces, hence E=1
2(6H5P). Moreover, each vertex is shared among three faces, hence
V=1
3(6H5P). Substituting the expressions for F,Eand Vinto eq. (1) yields P= 12.
These pentagons are the defects. Similarly, protein shells of spherical viruses which enclose
the genetic material consist of pentavalent and hexavalent subunits [6, 7]. Another condensed
3
matter analog of the geometrical frustration in footballs is the ‘colloidosome’. Colloidosomes
are spherical colloidal crystals [8–10] that are of considerable interest as microcapsules for
delivery and controlled release of drugs [8].
FIG. 1: Left panel: Geometric frustration in a football. A perfect tiling of hexagonal
panels cannot be achieved everywhere, resulting in black pentagonal panels (defects).
Right panel: Geometric frustration on the globe. The lines of latitude shrink to a point at
the north and south poles (defects). Adapted from
http://commons.wikimedia.org/wiki/File:Latitude_lines.
Another macroscopic example of geometrical frustration are the lines of latitude on the
surface of a globe. The points where these lines shrink to a point, that is the North and
South Poles, are the defects. Just like the pentagons on the football the defects on the globe
are also required by a topological constraint, namely the Poincare-Hopf theorem [5]:
X
a
sa=χ. (2)
The lines of latitude circle once around both poles. Hence, there are two defects with
an unit winding number, s. (See section V for a more precise definition.) Similar to the
lines of longitude and latitude on the globe, a coating of a nanoparticle with a monolayer
of ordered tilted molecules also has two polar defects [11–15]. Recently, Stellacci and co-
workers have been able to functionalise the defects to assemble linear chains of nanoparticles
[15]. A nematic liquid crystal coating possesses four defects at the vertices of a regular
tetrahedron in the ground state [12]. Attaching chemical linkers to these defects could
result in a three-dimensional diamond structure [13], rather than a one-dimensional chain.
This defect arrangement has been recently observed in nematic double emulsion droplets
4
[16], in which a nematic droplet itself contains another smaller water droplet. However, this
is only one of the many defect arrangements that are observed, as the size and location of
the inner water droplet is varied [16, 17]. Functionalisation of the defects, thus resulting
ordered structures confined to curved surfaces or shells offers an intriguing route to directed
assembly.
The types of order that we will discuss in this chapter are crystalline and (nematic) liquid
crystalline. After introducing mathematical preliminaries, we will discuss the elasticity of
crystals and liquid crystals and give a classification of the defects in these phases of matter.
We will elucidate the role of geometry in this subject. In particular, we will explicitly show
that, in contrast to the two examples given in the introduction, a topological constraint is
not necessary for geometrical frustration. After that we will explore the fascinating cou-
pling between defects and curvature. We will briefly comment on the screening by recently
observed charge-neutral pleats in curved colloidal crystals. We will then cross over from a
two dimensional surface to curved films with a finite thickness and variations in this thick-
ness. The particular system we are considering is a spherical nematic shell encapsulated
by a nematic double emulsion droplet. We will finish this chapter with a discussion on
nematic droplets of toroidal shape. Though topology does not prescribe any defects, there
is frustration due to the geometric confinement.
II. CRYSTALLINE SOLIDS AND LIQUID CRYSTALS
Besides the familiar solid, liquid and gas phases, there exist other fascinating forms
of matter, which display phenomena of order intermediate between conventional isotropic
fluids and crystalline solids. These are therefore called liquid crystalline or mesomorphic
phases [18, 19]. Let us consider the difference between a solid crystal and a liquid crystal.
In a solid crystal all the constituents are located in a periodic fashion, such that only
specific translations return the same lattice. Moreover, the bonds connecting neighbouring
crystal sites define a discrete set of vectors which are the same throughout the system. In
a crystal, there is thus both bond-orientational and translational order. In liquid crystals
there is orientational order, as the anisotropic constituent molecules define a direction in
space, but the translational order is partially or fully lost. The latter phase, in which
there is no translational order whatsoever, is called a nematic liquid crystal. The loss of
5
translational order is responsible for the fluidic properties of nematic liquid crystals. A
thorough introduction to liquid crystals can be found in the chapter by Lagerwall.
III. DIFFERENTIAL GEOMETRY OF SURFACES
A. Preliminaries
For a thorough introduction to the differential geometry of surfaces, please consult refs.
[3, 20, 21]. In this section we will introduce the topic briefly and establish the notation.
Points on a curved surface embedded in the three dimensional world we live in can be
described by a three-component vector R(x) as a function of the coordinates x= (x1, x2).
Vectors tangent to this surface are given by
tα=αR,(3)
where α=
∂xαis the partial derivative with respect to xα. These are in in general neither
normalised nor orthogonal. However, it does provide a basis to express an arbitrary tangent
vector nin:
n=nαtα.(4)
Here we have used the Einstein summation convention, i.e., an index occurring twice in a
single term is summed over, provided that one of the them is a lower (covariant) index and
the other is an upper (contravariant) index. We reserve Greek characters α,β,γ, . . . as
indices for components of vectors and tensors tangent to the surface. The so-called metric
tensor reads
gαβ =tα·tβ.(5)
and its inverse is defined by
gαβgβγ =δα
γ,(6)
where δα
γis equal to one if α=γand zero otherwise. We can lower and raise indices with
the metric tensor and inverse metric tensor, respectively, in the usual way, e.g.
gαβnα=nβ(7)
It is straightforward to see that the inner product between two vectors nand mis
n·m=nαtα·mβtβ=gαβnαmβ=nαmα.(8)
6
The area of the parallelogram generated by the infinitesimal vectors dx1t1and dx2t2, given
by the magnitude of their cross product, yields the area element
dS =
dx1t1×dx2t2
=q(t1×t2)2dx1dx2
=q|t1|2|t2|2(t1·t2)2dx1dx2
=g11g22 g12 g21dx1dx2
=gd2x(9)
where g= det(gαβ), the determinant of the metric tensor, and d2xis shorthand for dx1dx2.
More generally, the magnitude of the cross product of two vectors mand nis
|m×n|=
γαβmαnβ
,(10)
which introduces the antisymmetric tensor
γαβ =gαβ (11)
where αβ is the Levi-Civita symbol satisfying 12 =21 = 1 and is zero otherwise.
Since we will encounter tangent unit vectors, e.g. indicating the orientation of some
physical quantity, it is convenient to decompose this vector in a set of orthonormal tangent
vectors, e1(x) and e2(x), such that
ei·ej=δij and N·ei= 0,(12)
alternative to the basis defined in eq. 3. Here Nis the vector normal to the surface. We
use the Latin letters i,jand kfor the components of vectors expressed in this orthonormal
basis. As they are locally Cartesian they do not require any administration of the position
of the index. Besides the area element we need a generalisation of the partial derivative.
This generalisation is the covariant derivative, Dα, the projection of the derivative onto the
surface. The covariant derivative of nexpressed in the orthonormal basis reads in component
form [3]
Dαni=ei·αn
=ei·αnjej+ei·αejnj
=αni+ij Aαnj,(13)
7
where ij Aα=ei·αejis called the spin-connection. The final line is justified because
the derivative of any unit vector is perpendicular to this unit vector. More generally, the
covariant derivative of the vector nalong xαis [21]
Dαnβ=αnβ+ Γβ
αγ nγ(14)
where the Christoffel symbols are
Γα
βγ =1
2gαδ (γgβδ +βgδγ δgβ γ ).(15)
Finally, with the antisymmetric tensor and the area element in hand we can state a useful
formula in integral calculus, namely Stokes’ theorem
Zd2xαβ Dαnβ=Idxαnα.(16)
B. Curvature
The curvature is the deviation from flatness and therefore a measure of the rate of change
of the tangent vectors along the normal, or, put the other way around, a measure of the rate
of change of the normal along the tangent vectors. This can be cast in a curvature tensor
defined as
Kαβ =N·βtα.=tα·αN(17)
From this tensor we extract the intrinsic Gaussian curvature
G= det Kα
β=1
2γαβγγδ Kαβ Kγδ =κ1κ2(18)
and extrinsic mean curvature
H=1
2Tr Kα
β=1
2gαβKαβ =1
2(κ1+κ2),(19)
where κ1=N·1˜e1and κ2=N·2˜e2are the extremal or principal curvatures, the curvature
in the principal directions ˜e1and ˜e2. These eigenvalues and eigenvectors can be obtained
by diagonalising the matrix associated with the curvature tensor. If at a point on a surface
κ1and κ2have the same sign the Gaussian curvature is positive and from the outsiders’
point of view the surface curves away in the same direction whichever way you go, as is the
case on tops and in valleys. In contrast, if at a point on a surface κ1and κ2have opposite
8
FIG. 2: Saddle surface has negative Gaussian curvature. κ1and κ2have different signs.
Tangent circles are drawn in blue and red.
signs the Gaussian curvature is negative, the saddle-like surface curves away in opposite
directions. The magnitude of κ1and κ2is equal to the inverse of the radius of the tangent
circle in the principal direction (Fig. 2). It turns out that the Gaussian curvature and the
spin-connection are related. We will see how in a moment by considering the normal (third)
component of the curl (denoted by ∇×) of the spin-connection
(∇ × A)3=3jk j(e1·ke2)
=3jk je1·ke2
=3jk (N·je1) (N·ke2) (20)
where we have used the product rule and the antisymmetry of ijk in the second equality
sign. The final line is justified by the fact that the derivative of a unit vector is perpendicular
to itself and therefore we have e.g. je1= (N·je1)N+ (e2·je1)e2. If we now with the
aid of eqs. (18) and (17) note that
G= (N·1e1) (N·2e2)(N·1e2) (N·2e1) (21)
we easily see that the normal component of the curl of the spin-connection equals the
Gaussian curvature:
(∇ × A)·N=G, (22)
or alternatively[22]
γαβDαAβ=G. (23)
9
This geometrical interpretation of Awill show its importance in section IV, where we will
comment on its implications on the geometrical frustration in curved nematic liquid crystal
films.
C. Monge gauge
A popular choice of parametrisation of the surface is the Monge gauge or height represen-
tation in which x= (x, y) and R= (x, y, f (x, y)), where f(x, y) is the height of the surface
above the xy-plane. In this representation the Gaussian curvature reads
G=det αβf
g,(24)
where the determinant of the metric is given by
g= 1 + (xf)2+ (yf)2.(25)
IV. ELASTICITY ON CURVED SURFACES AND IN CONFINED GEOMETRIES
A. Elasticity of a two-dimensional nematic liquid crystal
In a nematic liquid crystal the molecules (assumed to be anisotropic) tend to align parallel
to a common axis. The direction of this axis is labeled with a unit vector, n, called the
director (see Fig. 3). The states nand nare equivalent. Any spatial distortion of a
n
FIG. 3: The director nspecifies the average local orientation of the nematic molecules.
uniform director field costs energy. If we assume that these deformations are small on the
molecular length scale, l,
|inj|  1
l,(26)
10
we can construct a phenomenological continuum theory. The resulting Frank free energy F
for a two dimensional flat nematic liquid crystal reads [18, 23, 24]
F=1
2Zd2xk1(ini)2+k3(ij inj)2,(27)
where the splay and bend elastic constants, k1and k3respectively, measure the energy of
the two independent distortions shown in Fig. 4. To simplify the equations one often makes
FIG. 4: Conformations with (left panel) a non-vanishing divergence of the director and
(right panel) a non-vanishing curl of the director.
the assumption of isotropic elasticity. In this approximation the Frank elastic constants are
equal, k1=k3=k, and up to boundary terms the free energy reduces to [23]
F=1
2kZd2x∂injinj,(28)
When the coupling of the director to the curvature tensor Kαβ [25–33] is ignored, the elastic
free energy on a curved surface generalises to [11, 13, 14, 34, 35]
F=1
2kZd2xgDαnβDαnβ,(29)
In this equation the area element has become dS =d2xgand partial derivatives have been
promoted to covariant derivatives. Because the director is of unit length, we can conveniently
specify it in terms of its angle with a local orthonormal reference frame, Θ (x), as follows
n= cos (Θ) e1+ sin (Θ) e2.(30)
Then, since αn1=sin (Θ) αΘ = n2αΘ and αn2= cos (Θ) αΘ = n1αΘ we see that
αni=ij njαΘ (31)
with which we find the covariant derivative to be
Dαni=ij nj(αΘAα) (32)
11
Therefore, we can rewrite the elastic energy as[22]
F=1
2kZd2xg(αΘAα) (αΘAα),(33)
where we have used that (ij nj) (ij nj) = δjk njnk= cos2(Θ) + sin2(Θ) = 1. This form
of the free energy [36] clearly shows that nematic order on curved surface is geometrically
frustrated. The topological constraints of the introductory section are merely a special
example of the frustration of local order due to the geometrical properties of the system.
Note that for a curved surface without such a topological constraint (e.g. a Gaussian bump)
the ground state can be a deformed director field. Since the curl of the spin-connection
equals the Gaussian curvature (eq. (23)), if the gaussian curvature is nonzero, the spin-
connection is irrotational and cannot be written as the gradient of a scalar field, Aα6=αΘ,
just like the magnetic field cannot be described by a scalar field either. Therefore Fin
eq. 33 is nonzero and we can conclude that there is geometrical frustration present in the
system.
B. Elasticity of a two-dimensional solid
Similar to the construction of the continuum elastic energy of a nematic liquid crystal, we
can write down the elastic energy of a linear elastic solid as an integral of terms quadratic
in the deformations, i.e. strain. This strain is found in the following way. Consider a
point x= (x, y, 0) on an initially flat solid. This point is displaced to x0(x)=(x0, y0, f ) in
the deformed solid, and so we may define a displacement vector u(x) = x0x=uxex+
uyey+fez. The square of the line element in the deformed plate is then given by ds02=
(dx +dux)2+ (dx +dux)2+df2. Noting that dux=iuxdxiwith xi=x, y and similarly for
uyand fwe find [37]
ds02=ds2+ 2uij dxidxj.(34)
Thus, the strain tensor uij (x) encodes how infinitesimal distances change in the deformed
body with respect to the resting state of the solid and reads
uij =1
2(iuj+jui+Aij ),(35)
where we have omitted non-linear terms of second order in iujand where the tensor field
Aij (x) is now defined as
Aij ifjf. (36)
12
We will assume that curvature plays its part only through this coupling of gradients of the
displacement field to the geometry of the surface, and we will therefore adopt the flat space
metric. This is a valid approximation for moderately curved solids, as we comment on at
the end of the section [38, 39]. To leading order in gradients of the height function, Aij is
related to the curvature as (see eq. (24))
1
2ikj lklAij = det (ijf) = G. (37)
Isotropy of the solid leaves two independent scalar combinations of uij that contribute to
the stretching energy: [37]
F=1
2ZdS 2µu2
ij +λu2
ii.(38)
The elastic constants λand µcalled the Lame coefficients. Minimisation of this energy with
respect to ujleads to the force balance equation:
iσij = 0,(39)
where the stress tensor σij (x) is defined by Hooke’s law
σij = 2µuij +λδij ukk.(40)
The force balance equation can be solved by introducing the Airy stress function, χ(x),
which satisfies
σij =ikj lklχ, (41)
since this automatically gives
iσij =jk k[1, ∂2]χ= 0 (42)
by the commutation of the partial derivatives. If one does not adopt the flat space metric,
the covariant generalisation of the force balance equation is not satisfied, because the the
commutator of the covariant derivatives, known as the Riemann curvature tensor, does not
vanish. It is actually proportional to the Gaussian curvature and indicates why the range of
validity of this approach is limited to moderately curved surfaces [38, 39]. Finally, for small
iujthe bond angle field, Θ (x), is given by
Θ = 1
2ij iuj.(43)
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C. Elasticity of a three-dimensional nematic liquid crystal
Besides splay and bend, there are two other deformations possible in a three dimensional
nematic liquid crystal. They are twist and saddle-splay, measured by elastic moduli K2and
K24. The analog of eq. (27) reads
F[n(x)] =1
2ZdV K1(∇ · n)2+K2(n·∇×n)2
+K3(n×∇×n)2K24 ZdS ·(n∇ · n+n×∇×n).
(44)
The integration of the splay, twist and bend energy density is over the volume to which the
nematic is confined. The saddle-splay energy per unit volume is a pure divergence term,
hence the saddle-splay energy can be written as the surface integral in eq. (44). In addition
to the energy in eq. (44), there is an energetic contribution coming from the interfacial
interactions, often larger in magnitude. Therefore, the anchoring of the nematic molecules
at the boundary can be taken as a constraint. In one of the possible anchoring conditions
the director is forced to be tangential to the surface, yet free to rotate in the plane. In this
case, the saddle-splay energy reduces to [40]
F24 =K24 ZdS κ1n2
1+κ2n2
2,(45)
thus coupling the director to the boundary surface. We refer to the chapter by Lagerwall
for a more detailed discussion on the origin of eq. (44).
V. TOPOLOGICAL DEFECTS
Topological defects are characterised by a small region where the order is nod defined.
Topological defects in translationally ordered media, such as crystals, are called dislocations.
Defects in the orientational order, such as in nematic liquid crystals and again crystals, are
called disclinations. The defects are topological when they cannot be removed by a contin-
uous deformation of the order parameter. As we will see momentarily, they are classified
according to a topological quantum number or topological charge, a quantity that may only
take on a discrete set of values and which can be measured on any circuit surrounding the
defect.
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A. Disclinations in a nematic
Consider for concreteness a two dimensional nematic liquid crystal. A singularity in
the director field is an example of a disclination. Such a point defect can be classified by
its winding number, strength, or topological charge, s, which is the number of times the
director rotates by 2π, when following one closed loop in counterclockwise direction around
the singularity:
IdΘ = IdxααΘ = 2πs (46)
We can express eq. (46) in differential form by invoking Stokes’ theorem:
γαβDαβΘ = (xxa) (47)
where we use an alternative labelling, q= 2πs, of the charge of the defect, which is located
at xa. The delta-function obeys
δ(xxa) = δ(x1x1
a)δ(x2x2
a)
g,(48)
such that the integral over the surface yields one. The far field contribution of the defect to
the angular director in a flat plane reads
Θ = +c, (49)
as it forms a solution to the Euler-Lagrange equation of the elastic free energy
2Θ = 0.(50)
Here, φis the azimuthal angle and cis just a phase. Examples are presented in Fig. 5. Note
that since the states nand nare equivalent, defects with half-integer strength are also
possible. In fact, it is energetically favourable for an s= 1 defect to unbind into two s=1
2
defects [13, 41].
B. Disclinations in a crystal
Though energetically more costly, disclinations also arise in two-dimensional crystals. At
these points the coordination number deviates from its ordinary value, which is six for a
crystal on a triangular lattice. Just like in nematic liquid crystals, disclinations in crystals are
15
(a) s= 1, c= 0 (b) s= 1, c=π
4(c) s= 1, c=π
2(d) s=1
2,c= 0
FIG. 5: Director configurations, n1= cos Φ, n2= sinΦ, for disclinations of strength sand
constant c.
labelled by a topological charge, q, which is the angle over which the vectors specifying the
lattice directions rotate when following a counterclockwise circuit around the disclination. If
we parametrise these lattice direction vectors with Θ(x), the bond-angle field, this condition
reads mathematically
IdΘ = q. (51)
Thus for disclinations in a triangular lattice with five-fold and seven-fold symmetry, as
displayed in Fig. 6, q=π
3and q=π
3respectively. Analogous to eq. (47), the flat-space
FIG. 6: (Left panel) Five-fold and (right panel) seven-fold disclination. When following a
closed counterclockwise loop (red) around the five-fold disclination, the initial lattice
vector a1rotates via a2,a3,a4and a5over an angle of π/3 to a6.
differential form of eq. (51) for a disclination located at xareads
ij ijΘ = (xxa) (52)
16
C. Dislocations
Besides disclinations, dislocations can occur in crystals. Dislocations are characterised
by a Burger’s vector b. This vector measures the change in the displacement vector, if we
make a counterclockwise loop surrounding the dislocation,
Idu=b.(53)
Just like the strength of disclinations can only take on a value out of a discrete set, the
Burger’s vector of a dislocation is equal to some integer multiple of a lattice vector. Also
note that a dislocation can be viewed as a pair of closely spaced disclinations of opposite
charge [42], as can be seen in Fig. 7.
FIG. 7: Dislocation in a triangular lattice. The Burger’s vector specifies by how much a
clockwise circuit (marked in red, bold) around the dislocation fails to close. A dislocation
can be viewed as disclination dipole with a moment perpendicular to its Burger’s vector.
The flat space differential form of eq. (53) for a dislocation at xais
ij ijuk=bkδ(xxa),(54)
which again can be obtained by using Stokes’ theorem.
VI. INTERACTION BETWEEN CURVATURE AND DEFECTS
A. Coupling in liquid crystals
It is possible to recast the free energy in terms of the locations of the topological defects
rather than the director or displacement field, if smooth (i.e. non-singular) deformations
17
are ignored. In this case the energy in eq. (33) is minimised with respect to Θ, leads to
Dα(αΘAα) = 0.(55)
This needs to supplemented with an equation for the effective charge distribution:
γαβDα(βΘAβ) = ρG, (56)
obtained by combining eq. (23) for the curvature and eq. (47) for the defect density ρ(x),
ρ=X
a
qaδ(xxa).(57)
Eq. 55 is automatically satisfied if one chooses [35]
αΘAα=γβ
αβχ, (58)
where χ(x) is an auxiliary function. At the same time, substituting eq. (58) into eq. (56)
leads to
D2χ=ρG. (59)
The source in this Poisson equation contains both topological point charges as well as the
Gaussian curvature with opposite sign. The analog of the electrostatic potential is χ. The
role of the electric field is played by αχ. Indeed, substituting eq. (58) in eq. (33), shows
that the energy density is proportional to the square of the electric field:
F=1
2kZdS∂αχ∂αχ. (60)
Next, we formally solve eq. (59)
χ=ZdS0ΓL(x,x0) (ρ(x0)G(x0)) (61)
where ΓL(x,x0) is the Green function of the Laplace-Beltrami operator, D2=DαDα, satis-
fying
D2ΓL(x,x0) = δ(xx0).(62)
Integrating eq.(60) by parts and substituting our expressions for χand the Laplacian of χ
(eqs. (61) and (59) respectively) results (up to boundary terms) in
F=k
2ZdS ZdS0(ρ(x)G(x)) ΓL(x,x0) (ρ(x0)G(x0)) ,(63)
18
from which we again deduce the analogy with two-dimensional electrostatics. In this analogy
the defects are electric point sources with their electric charge equal to the topological charge
qand the Gaussian curvature with its sign reversed is a background charge distribution.
Therefore the defects will be attracted towards regions of Gaussian curvature with the
same sign as the topological charge [11, 30, 35, 43–47]. Such screening will be perfect if
S=ρeverywhere, since F= 0 then. However, unless the surface contains singularities
in the Gaussian curvature, like the apex of a cone, perfect screening will be impossible, as
the topological charge is quantised whereas the Gaussian curvature is typically smoothly
distributed.
B. Coupling in crystals
Note that an arbitrary field χsolves eq. (42). However, χmust be physically possible
and we therefore need to accompany eq. ((42)) with another equation, which we will obtain
by considering the inversion of eq. (40) [37, 48]:
uij =1 + ν
Yσij ν
Yσkkδij (64)
=1 + ν
Yikj lklχν
Y2χδij (65)
where the two-dimensional Young’s modulus, Y, and Poisson ratio, ν, are given by
Y=4µ(µ+λ)
2µ+λ,(66)
ν=λ
2µ+λ.(67)
Applying ikj lklto eq. (65) gives
1
Y4χ=ikj lkluij .(68)
By invoking eqs. (35), (43), the differential expressions for the defects, namely eqs. (54)
and (52), as well as eq. (37) for the curvature, one can rewrite the right hand side to arrive
at the crystalline analog of eq. (59):
1
Y4χ=ρG, (69)
where the defect distribution, ρ, of disclinations with charge qaand dislocations with Burger’s
vector bbreads
ρ=X
a
qaδ(xxa) + X
b
ij bb
ijδ(xxb).(70)
19
We can also rewrite the free energy (up to boundary terms) in terms of the Airy stress
function as follows:
F=1
2YZdS 2χ2(71)
If we integrate this by parts twice and use eq. (69) to eliminate χand 4χ, we find (up to
boundary terms)
F=Y
2ZdS ZdS0(ρ(x)G(x)) ΓB(x,x0) (ρ(x0)G(x0)) (72)
where ΓBis the Greens function of the biharmonic operator
4ΓB(x,x0) = δ(xx0).(73)
Eq. (72) is the crystalline analog of eq. (63). Again, the defects can screen the Gaussian
curvature. The interaction, however, is different than the Coulomb interaction in the liquid
crystalline case. If the surface is allowed to bend, disclinations will induce buckling, illus-
trated in Fig. 8 with paper models. In these cones, the integrated Gaussian curvature is
determined by the angular deficit of the disclination
ZdSG =q. (74)
C. Screening by dislocations and pleats
Surprisingly, also charge neutral dislocations and pleats can screen the curvature [10,
22, 38, 48, 49]. Pleats are formed by arrays of dislocations and allow for an extra piece of
crystal, just like their fabric analogs. The opening angle, ∆Θ, of the pleat (or low angle
grain boundary) is given by
∆Θ nda(75)
where ais the lattice spacing and ndis the dislocation line density. Since this opening angle
can be arbitrarily small, pleats can provide a finer screening than quantised disclinations.
D. Geometrical potentials and forces
The cross terms of equation (72) represent the interaction energy
ζ=YZdSρ (x)ZdA0ΓB(x,x0)G(x0) (76)
20
FIG. 8: Paper models illustrating the coupling between disclinations and curvature. Left
panels: Positively (right panels: negatively) charged disclinations and positive (negative)
Gaussian curvature attract. Top left panel: 5-fold coordinated particle in a triangular
lattice. Top right panel: 7-fold coordinated particle in a triangular lattice. Bottom left
panel: 3-fold coordinated particle in a square lattice. Bottom right panel: 5-fold
coordinated particle in a square lattice.
By introducing an auxiliary function V(x) satisfying
2V=G, (77)
eq. (76) can by integrating by parts twice be rewritten as
ζ=YZdSρ (x)ZdS0ΓL(x,x0)V(x0) (78)
The field ζ(x) can be viewed as a geometric potential, i.e. the potential experienced by a
defect due to the curvature of the crystal [38, 39]. Another, more heuristic way, to study
the interaction of dislocations and curvature is the following. We consider the stress that
exist in the monolayer as a result of curvature only, σG
ij , as the source of a Peach-Koehler
21
force, f, on the dislocation:
fk=kj biσG
ij .(79)
Note that, by setting ρ= 0, the Airy stress function χGsatisfies
1
Y4χG=G(80)
This equation can be solved in two steps. First, we make use of an auxiliary function U
obeying
2U=G(81)
This leaves the following equation to be solved
1
Y2χG=U+UH,(82)
where UHis a harmonic function (i.e. 2UH= 0) introduced to fulfil the boundary conditions
[38].
VII. NEMATICS IN SPHERICAL GEOMETRIES
A. Nematic order on the sphere
As a naive guess for the ground state of a two dimensional nematic liquid crystal phase on
the surface of the sphere, one could imagine the excess of topological charge to be located at
the poles, like in the case of tilted molecules on the sphere. However, the order parameter,
the director, has the symmetry of a headless arrow instead of a vector. Therefore, this
makes it possible for the two s= 1 defects to unbind into four s=1
2defects relaxing at the
vertices of a regular tetrahedron [12]. The baseball-like nematic texture is illustrated in Fig.
9. The repulsive nature of defects with like charges can be seen from the free energy, which,
as shown in the previous section, can entirely be reformulated in terms of the defects rather
than the director [12, 13]:
F=πk
2X
i6=j
sisjlog (1 cos θij ) + E(R)X
j
s2
j.(83)
Here, θij is the angular separation between defects iand j, i.e. θij =dij
R, with dij being
the geodesic distance. The first term yields the long-range interaction of the charges. The
22
FIG. 9: The baseball-like ground state of a two-dimensional spherical nematic coating has
four s=1
2at the vertices of a tetrahedron in the one-constant approximation. Figure from
[14].
second term accounts for the defect self-energy
E(R) = πk log R
b+Ec,(84)
where we have imposed a cut-off brepresenting the defect core size, which has energy Ec.
This cut-off needs to be introduced in order to prevent the free energy from diverging.
Heuristically, this logarithmically diverging term in the free energy is responsible for the
splitting of the two s= 1 defects into four s=1
2defects. Two s= 1 defects contribute
(2 ×12)πk log R
b= 2πk log R
bto the free energy, whereas four s=1
2defects contribute
only 4×1
22πk log R
b=πk log R
b.
In addition to this ground state, other defect structures have been observed in computer
simulations [50–53]. If there is a strong anisotropy in the elastic moduli, the four defects are
found to lie on a great circle rather than the vertices of a regular tetrahedron [52, 53].
B. Beyond two dimensions: spherical nematic shells
An experimental model system of spherical nematics are nematic double emulsion droplets
[16, 17, 54–60]. These are structures in which a water droplet is captured by a larger nematic
liquid crystal droplet, which in turn is dispersed in an outer fluid. There are some crucial
differences between a two-dimensional spherical nematic and these systems. Not only is the
23
nematic coating of a finite thickness, this thickness can be inhomogeneous as a result of
buoyancy driven displacement (or other mechanisms) of the inner droplet out of the centre
of the nematic droplet.
Like point disclinations in two dimensions, there exist disclination lines in a three di-
mensional nematic liquid crystal, which are categorised in similar fashion. However, charge
one lines, and integral lines in general, do not exist. Such lines loose their singular cores
[61, 62] by ‘escaping in the third dimension’. In shells, such an escape leads to another type
of defects, namely point defects at the interface, known as boojums (Fig. 10).
FIG. 10: (a) Schematic of the deconfined defect configuration in a homogeneous shell. Two
pairs (each encircled in red) of boojums, indicated by green dots, are located at the top
and bottom of the shell. (b) Schematic of the confined defect configuration in an
inhomogeneous shell. All boojums are located at the thinnest, top part of the shell, inside
the red rectangle. (c) Zoom of the thinnest section of the inhomogeneous shell in (b).
From [17] - Reproduced by permission of The Royal Society of Chemistry.
In a spherical nematic layer of finite thickness, calculations show that the baseball struc-
ture with four s=1
2disclination lines spanning the shell, become energetically less favourable
than two antipodal pairs of boojums beyond a critical thickness [14]. Instead of unbind-
24
ing, the singular lines escape in the third dimension, leaving two pairs of boojums on the
bounding surfaces. These two defect configurations are separated by a large energy barrier.
As a consequence, both configurations are observed in droplets in the same emulsion. If,
in addition, the shell thickness is inhomogeneous, the energy landscape becomes even more
complex.
As a consequence of the inhomogeneity the defects cluster in the thinnest part of the
shell, where the length of the disclination lines (or distance between boojums forming a
pair) are shorter. Since the self-energy of the disclination is proportional to its length,
it is attracted towards this region of the shell. One of the intriguing outcomes of the
study of inhomogeneous shells is that in the two defects shell, the pairs of surface defects
can make abrupt transitions between the state in which the defects are confined in the
thinnest part of the shell, and the deconfined state, in which the interdefect repulsion places
them diametrically [17]. These confinement and deconfinement transitions occur when the
thickness or thickness inhomogeneity is varied. A defect arrangement with a corresponding
local minimum in the energy landscape makes the transition to the global minimum when
the local minimum looses its stability. This explains both the abruptness of the transitions
as well as the hysteresis between them.
In agreement with this picture, Monte Carlo simulations of nematic shells on uniaxial
and biaxial colloidal particles have shown the tendencies for defects to accumulate in the
thinnest part and in regions of the highest curvature [63].
VIII. TOROIDAL NEMATICS
The torus has a zero topological charge. Hence, in a nematic droplet of toroidal shape no
defects need to be present. The director field to be expected naively in such a geometry is
one which follows the tubular axis, as shown in Fig. 11. This achiral director configuration
contains only bend energy. Simple analytical calculations show, however, that if the toroid
becomes too fat it is favourable to reduce bend deformations by twisting. The price of
twisting is screened by saddle-splay deformations provided that K24 >0 [40, 64]. The twisted
configuration is chiral. Chirality stems from the Greek word for hand, and is indeed in this
context easily explained: your right hand cannot be turned into a left hand by moving and
rotating it. It is only when viewed in the mirror that your right hand appears to be a left hand
25
and vica versa. Indeed, for small aspect ratios and small values of (K2K24)/K3nematic
toroids display either a right- or left-handedness despite the achiral nature of nematics. This
phenomenon is recognised as spontaneous chiral symmetry breaking. Typical corresponding
plots of the energy as a function of the amount of twist are shown in Fig. 11.
FIG. 11: Energy as a function of the degree of twist has either a single achiral minimum
(dashed blue) or shows spontaneous chiral symmetry breaking in toroidal nematics (red)
depending on the aspect ratio and elastic constants. The chiral state is favoured for fat
toroids and small values of (K2K24)/K3.
IX. CONCLUDING REMARKS
We hope to have shared our interest in the rich subject of geometry in soft matter, in
particular the interplay of defects and curvature in two-dimensional ordered matter and the
confinement of nematic liquid crystals in various geometries. For readers interested in a
more detailed treatment, we refer to excellent reviews by Kamien[3], Bowick and Giomi[39],
26
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