Topological Phases and Curvature-Driven Pattern Formation in Cholesteric Shells G. Negro1 L.N. Carenza2 G. Gonnella3 D. Marenduzzo1 and E. Orlandini4 1SUPA School of Physics and Astronomy University of Edinburgh

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Topological Phases and Curvature-Driven Pattern Formation in Cholesteric Shells

G. Negro1, L.N. Carenza2, G. Gonnella3, D. Marenduzzo1, and E. Orlandini4∗

1SUPA, School of Physics and Astronomy, University of Edinburgh,

Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK

2Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

3Dipartimento di Fisica, Università degli Studi di Bari and INFN,

Sezione di Bari, via Amendola 173, Bari, I-70126, Italy. and

4Dipartimento di Fisica e Astronomia, Università di Padova, 35131 Padova, Italy

(Dated: February 7, 2023)

We study the phase behaviour of cholesteric liquid crystal shells with diﬀerent geometries. We

compare the cases of tangential and no anchoring at the surface, focussing on the former case, which

leads to a competition between the intrinsic tendency of the cholesteric to twist and the anchoring

free energy which suppresses it.We then characterise the topological phases arising close to the

isotropic-cholesteric transition.These typically consist of quasi-crystalline or amorphous tessellations

of the surface by half-skyrmions, which are stable at lower and larger shell size respectively. For

ellipsoidal shells, defects in the tesselation couple to local curvature, and according to the shell size

they either migrate to the poles or distribute uniformly on the surface. For toroidal shells, the

variations in the local curvature of the surface stabilises heterogeneous phases where cholesteric or

isotropic patterns coexist with hexagonal lattices of half-skyrmions.

I. INTRODUCTION

Cholesteric liquid crystals (CLC) are fascinating soft

materials that naturally exhibit a wide zoology of topo-

logical phases and singularities –the so-called topologi-

cal defects–and therefore they provide a proliﬁc ground

for testing abstract topological theories in many areas of

physics and mathematics [1–5]. The origin of the inter-

est of the scientiﬁc community however, is due to vari-

ous striking optical properties of CLC. For instance, they

have a chiral pitch of the same order of magnitude as

the wavelength of visible light, respond to applied elec-

tromagnetic ﬁelds and exhibit important birefringence

properties. Therefore they are extremely valuable in a

wealth of applications to optical devices, liquid crystal-

based lasers and nanotechnology [6–9], as well as in bio-

inspired realizations [10–12].

Among others, Blue Phases (BP) are examples of CLC

that occur in proximity of the isotropic-nematic transi-

tion and manifest themselves as a network of double twist

cylinders that self-assemble into three-dimensional (3D)

structures [1,13–15], which appear in vivid shades of blue

– to which they owe their name. In 3D BPs come in three

diﬀerent forms. BPI and BPII feature cubic arrange-

ments in the network of disclination lines and therefore

have a crystalline nature, and ﬁnd industrial applications

in designing fast light modulators [16], lasers [6], display

devices [17] and tunable photonic crystals [18]. Instead,

BPIII still features a network of disclination lines [19],

but arranged in a disordered fashion to yield an amor-

phous structure [19,20], and it is also referred to as the

blue fog, with the name deriving from a loss in the ob-

served brightness of the coloured texture.

∗carenza@lorentz.leidenuniv.nl

CLC –and BPs in particular– are fully 3D structures

and, when conﬁned within restricted regions, competition

between geometrical constraints and 3D defects gives rise

to a phenomenon known as topological frustration that

may produce a plethora of new defect arrangements or

topological phases [21–24]. Within quasi-2D CLC sam-

ples, the chirality of the molecules favours the onset of

double-twisted cylinders, which can be viewed as topolog-

ical quasi-particles known as half-skyrmions (or merons).

In other quasi-2D geometries, even more exotic structures

such as knotted and linked disclination loops (hopﬁons)

can be observed [8,25,26].

The emerging physics becomes even richer when the

CLC are conﬁned in a region bordered by curved surfaces.

Such cases can be experimentally obtained by encapsu-

lating a small amount of liquid crystal within spherical

shells [27]. Conﬁned nematic LC [28,29] yield a panoply

of diﬀerent conﬁgurations were observed depending on

the alignment (homeotropic or tangential) and thickness

of the liquid crystalline shell. For instance, in the case

of tangential anchoring, topological conﬁnement follows

from the Gauss-Bonnet theorem, stating that the total

topological charge (the sum of the charge of all defects)

must equal the Euler characteristic of the bounding ge-

ometry. Therefore, for spherical shells this requires a

total s= +2 defective charge which can be realised ei-

ther with four +1/2disclinations, or with two boojums,

each carrying +1 topological charge, or even with more

complex states [27]. Very recently this technique has

been extended to CLC [23] which lead to the additional

appearance of skyrmions, merons and other transitory

states, thanks to the competition between the anchoring

direction of the LC molecules at the boundary, the sur-

face curvature and the cholesteric pitch of the LC, which

introduces yet an additional length-scale.

Notwithstanding this initial experimental evidence, lit-

tle is known from a theoretical perspective on the order-

arXiv:2210.04877v2 [cond-mat.soft] 6 Feb 2023

ing properties of CLC conﬁned in non-ﬂat geometries. To

ﬁll this gap, in a recent paper [22] we made use of the

Landau-de Gennes ﬁeld theory to study the equilibrium

properties of CLC conﬁned into shells (cholesteric shells).

By means of lattice Boltzmann simulations, we found

that cholesteric shells support various types of topolog-

ical phases that are completely absent on the quasi-2D

ﬂat geometries, analysed for instance in [30].

In particular, under strong geometrical conﬁnement

obtained by placing the CLC on a shell with a radius

comparable with the cholesteric pitch, quasi-crystalline

lattices of half-skyrmions were observed with several pos-

sible mixtures of polygonal tessellations, depending both

on the radius and the chirality of the CLC. Interestingly,

yet another phase appeared at larger chirality, with the

loss of quasi-crystalline order leading to an amorphous

patching of half-skyrmions. This may be viewed as a 2D

counterpart of the 3D amorphous blue phase, the blue

fog. These results are especially notable if one considers

that this wide zoology of possible states were observed

in absence of anchoring of the LC molecules on the shell

surface. Anchoring is indeed expected to play a signiﬁ-

cant role on the stability and degree of ordering of the

topological phases discovered in [22]: for instance, a ﬁnite

anchoring on a thin shell it should suppress twisting of

the director ﬁeld, which is thermodynamically favoured

in CLC. Moreover, studying the stability of topologi-

cal phases in presence of perturbing interactions, as an-

choring, is relevant even from an experimental perspec-

tive as liquid crystalline shells can be stabilized in the

lab through surfactants which often favour a preferen-

tial orientation of liquid crystal molecules at the inter-

face [31,32]. Another potentially important issue which

is poorly investigated is the eﬀect of curvature on the

half-skyrmion organisation and on topological phases in

general. This is a relevant issue to explore given that soft

shells can easily be deformed locally, thereby providing a

way to control local curvature experimentally.

In this article we will address these two open ques-

tions. First, we will consider the case of a CLC spherical

shell with tangential anchoring at the surface and we will

show how an orientational energetic penalty can signiﬁ-

cantly stabilize novel quasi-crystalline structures by hin-

dering the proliferation of defects in the tessellation. Fur-

thermore, we will show how geometries with non-uniform

Gaussian curvature can be exploited to control defect po-

sitions, by considering both ellipsoidal and toroidal shells.

The article is organized as follows. In Section II

we provide details of Landau-de Gennes free-energy of

cholesteric shells with tangential anchoring and their as-

sociated dynamical equations. In Section III we deter-

mine the equilibrium phase diagram of the system and

characterize the order and regularity of the discovered

topological phases in terms of both a generalised hex-

atic order parameter and the Bessel structure factor. To

pinpoint the role of the anchoring we compare these re-

sults with those previously obtained in absence of LC

anchoring at the surface [22]. In Section IV we extend

our analysis to ellipsoidal and toroidal cholesteric shells.

The goal here is to understand how the presence of a

non-homogeneous local curvature impacts on the stabil-

ity and the patterning of the observed structures. We

will ﬁnally discuss the results in Section V.

II. MODEL

We start by introducing the relevant order parameters

for describing the state of the system. The orientational

properties of the liquid crystal (LC) are customarily de-

scribed by the symmetric and traceless tensor Q(r). The

principle direction of the tensor, deﬁned as the eigen-

vector nof the largest eigenvalue, represents the local

alignment direction of the liquid crystal molecules in a

certain point in space, while the amplitude pTr(Q2)is

a measure of the degree of liquid crystalline order. In

the following we will identify topological defects on the

shell surface by measuring the degree of biaxiality of the

LC, looking at the second parameter of the Westin met-

rics cp[33] [34]. The phase ﬁeld, which determines the

geometry on which we will conﬁne the liquid crystal, is

deﬁned by the scalar ﬁeld φ(r). Finally, the velocity ﬁeld

vcaptures the local velocity of the liquid phase.

II.1. Free Energy

The ground state of the system is deﬁned by the free

energy

FQ=ZdrA01

21−χ

3Q2−χ

3Q3+χ

4Q4

2(∇ · Q)2+∇ × Q+ 2q0Q)2.(1)

Here, the terms on the ﬁrst line, proportional to the bulk

constant A0, capture the ﬁrst-order isotropic-nematic

transition. This occurs when the temperature-like pa-

rameter χ > χcr = 2.7. The terms on the second line,

proportional to the elastic constant L, are the elastic con-

tributions to the free energy and account for the elastic

energy penalty due to LC deformation. For non-zero val-

ues of the parameter q0, the mirror symmetry is bro-

ken and chiral states emerge at equilibrium. In particu-

lar, in the following, we will consider the case of right-

handed twist (q0>0) with the equilibrium pitch given

by p0= 2π/q0in bulk systems. It is worth noticing that

the saddle splay term of the director ﬁeld theory is in-

herently eﬀectively incorporated in our model, as shown

in[35].

The phase ﬁeld ground state is determined by minimis-

ing the following free energy:

Fφ=Zdra

4φ2(φ−φ0)2+kφ

2(∇φ)2.

For a > 0, the polynomial term has two possible equi-

librium values (minima) at φ= 0, φ0. The two mate-

rial parameters aand kφaccount for the surface tension

γ=p8akφ/9and the interface width ξφ=p2kφ/a.

To create and stabilise a LC shell we require the

temperature-like parameter χto depend on the gradients

of the phase ﬁeld φas follows

χ=χ0+χs(∇φ)2.

In this way the Q-tensor attains non-zero values only in

those regions where |∇φ|>p(χcr −χ0)/χs(see Ap-

pendix V). For spherical droplets, φis φ0inside the

droplet and 0otherwise, so that the square of the gra-

dient (∇φ)2is maximal at the interface, and approaches

0suﬃciently far from it. In our model χdepends on

(∇φ)2in such a way that the system is liquid crystalline

at the interface and isotropic away from it, resulting in a

thin shell of liquid crystal with isotropic ﬂuid inside and

outside. Diﬀerent (non-spherical) shell geometries can be

obtained by letting φ0attain a spatial modulation:

φ0(r) = (¯

φ0if T(r)<0

0otherwise (2)

where ¯

φ0is a constant and T(r)is the parametric

equation of a generic manifold. For instance, in the

case of a torus with radii R1and R2(with R1> R2)

T(x, y, z) = (R1−(x2+y2)1/2)2+z2−R2

2. There-

fore, the interface is formed where T ≈ 0, hence on the

torus surface. Analogously, for the case of an ellipsoid

T(x, y, z) = x2/R2

1+y2/R2

2+z2/R2

3−1, being R1, R2, R3

the three semiaxis of the ellipsoid. This approach is ap-

propriate for cases where we are not interested in defor-

mations or motion of the shell, such as ours. Notice that

the phase ﬁeld free energy in Eq. (II.1), gives a descrip-

tion of the interface equivalent to that of the Helfrich

model (see e.g. [36]). The gradient square term in the

coupling with the liquid crystal is equivalent to a delta

function selecting points at the interface. This serves as

a regularization in the context of phase ﬁeld models [37].

Finally, soft anchoring at the surface of the shell is

imposed through an additional coupling term in the free

energy involving the Q-tensor and the gradient of the

phase ﬁeld,

Fanch =Zdrβ[∇φ·Q· ∇φ].(3)

Tangential anchoring is achieved by choosing β > 0: we

focus on such a case in this work.

II.2. Dimensionless numbers

The dimensionless numbers determining the behaviour

of the system are as follows: (i) the reduced tem-

perature τ= 9(3 −χ)/χ,(ii) the chirality strength

κ=p108q2

0L/(A0χ)proportional to the ratio between

cholesteric pitch and nematic coherence length ξn=

pL/A0[1,14], and (iii) the ratio between shell ra-

dius and cholesteric pitch, R/p0[21].Moreover, we choose

parameters so that the interface thickness ξφp0.

In fully 3D systems and low chirality (κ0.5) for

τ < τc(κ) = 1

2h1−4κ2+1+4κ2/33/2i, Eq. (1) is

minimised by the helical phase [38] that manifests itself

as unidirectional twisting of liquid crystalline molecules

and with equilibrium pitch p0= 2πq−1

0. As chirality

grows larger (κ0.5), double twist of the cholesteric

helix becomes favorable and a new metastable phase–

the blue phase–appears. In the mean-ﬁeld approach of

Ref. [38], for large values of the reduced chirality κ, the

system is isotropic when τ > 0.8τc(κ= 0), it develops

blue phases when the reduced temperature is comprised

between 0.8τc(κ= 0) and τc(κ), and is in its helical phase

for τ < τc(κ). In 2D, blue phases appear as a regular half-

skyrmion lattice with hexagonal symmetry. A numerical

phase diagram of the stability of such phases in 2D is

provided in [25,30].

II.3. Dynamical Equations

Nemato-hydrodynamics is ruled by the incompressible

Navier-Stokes equation for the ﬂow ﬁeld vand the Beris-

Edwards equation for the Q-tensor. The former is given

ρ(∂tv+v· ∇v) = −∇p+∇ · (σvisc +σel),

where ρis the total (constant) density of the ﬂuid and p

is the hydrodynamic pressure. The viscous contribution

to the stress tensor is given by

σvisc

αβ =η(∂αvβ+∂βvα)

where ηis the nominal viscosity of the isotropic ﬂuid.

The elastic stress

σel

αβ =−˜

ξHαγ Qγβ +1

3δγβ −˜

ξQαγ +1

3δαγ Hγβ

+ 2˜

ξQαβ −1

3δαβQγµHγµ +Qαγ Hγβ −Hαγ Qγβ

−∂αQγν

∂f

∂∂βQγν

(4)

is responsible for the backﬂow that originates from de-

formations in the LC arrangement. [In the equation

above, fdenotes the free energy density correspond-

ing to the free energy FQ+Fφ+Fanch.] Here, H=

−δF

δQ+I

3T r δF

δQis the molecular ﬁeld (with F=

FQ+Fφ+Fanch the total free energy), fis the free

energy density corresponding to F, while ˜

ξis the ﬂow-

alignment parameter which controls the aspect-ratio of

the liquid crystal molecules and aligning properties to

OHS

OHP

AMORPHOUS

OHP

AMORPHOUS

Figure 1. Phase behaviour and typical shell conﬁgurations with tangential anchoring. Panel (a) shows the phase

diagram in the κ−R/p0plane for spherical shells with tangential anchoring (β= 8 ×10−3). In the legend, S=square,

P=pentagons, H=hexagons, O=octagons, HE=heptagons. Panel (b) shows the pattern of director ﬁeld of half-skyrmions and

−1/2defects deﬁning octagons, hexagons and pentagons for a magniﬁed region of the conﬁguration in panel (d). Three quasi-

crystalline conﬁgurations are shown in panels (c-e), for the same values of reduced chirality κ= 0.6(or equivalently L= 0.02)

and varying values of the radius R. These are: (c) HS at R/p0= 1 (R= 25), (d) OHS at R/p0= 1.2(R= 30) and (e) OHP at

R/p0= 2 (R= 50). Panel (g) shows an amorphous conﬁguration at R/p0= 3.2(R= 80), for the same value of reduced chirality

κ= 0.6as previous cases. The dashed line in panel (a) highlights the region with constant κwhere the conﬁgurations (c-f)

were taken. The color code in panel (b-f) corresponds to the isotropic parameter csof the Westin metrix [33]: red-white regions

deﬁne defect positions (Q∼0), while blue ones are ordered. Panels (f) and (h) show the Schlieren textures corresponding to

the OHP and amorphous conﬁgurations, respectively. The stability of the observed conﬁgurations has been tested on several

runs for a selected number of parameter values by varying the random initial conditions and ﬁxing the physical parameters of

the simulation.

the ﬂow (we chose ˜

ξ= 0.7to consider ﬂow-aligning rod-

like molecules).

The Beris-Edwards equation for Q-tensor evolution is

given by

(∂t+v· ∇)Q− S(∇v,Q)=ΓH,

where Γis the rotational viscosity which measures the

relative importance of advective interactions with respect

of the driving eﬀect of the molecular ﬁeld Hand

S(∇v,Q)=(˜

ξD+Ω)(Q+I/3) + (Q+I/3)(˜

ξD−Ω)

−2˜

ξ(Q+I/3)T r(QW)

(5)

is the strain-rotational derivative. Here, Dis the strain-

rate tensor and Ωthe vorticity tensor, respectively given

by the symmetric and anti-symmetric part of the velocity

gradient tensor W=∇v=∂βvα.

Finally, in this paper we are not interested in the for-

mation process of spherical shells but rather on the re-

laxation of the liquid crystal on curved geometries. This

is equivalent to requiring that the typical relaxation time

of the liquid crystal dynamics are long compared with

those of the phase ﬁeld. Therefore, after thermalization,

we assume the phase ﬁeld φas a static non-evolving ﬁeld

and its conﬁguration is obtained by a free energy energy

minimisation procedure for each geometry considered.

III. SPHERICAL SHELLS: EQUILIBRIUM

PHASE DIAGRAM AND ANCHORING EFFECTS

We start our discussion by considering spherical shells.

We ﬁx the reduced temperature τ= 0.540 and the

cholesteric pitch p0= 25.64. We enforce soft tangential

anchoring by setting the anchoring constant β= 8×10−3

and look for the equilibrium states obtained by varying

values of the chirality strength κand of the ratio between

the shell radius and the helix pitch, R/p0. These are in

turn tuned by varying respectively the elastic constant L

and, for spherical shells, their radius R.

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TopologicalPhasesandCurvature-DrivenPatternFormationinCholestericShellsG.Negro1,L.N.Carenza2,G.Gonnella3,D.Marenduzzo1,andE.Orlandini41SUPA,SchoolofPhysicsandAstronomy,UniversityofEdinburgh,PeterGuthrieTaitRoad,Edinburgh,EH93FD,UK2Instituut-Lorentz,UniversiteitLeiden,P.O.Box9506,2300RALeiden,TheNet...

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Topological Phases and Curvature-Driven Pattern Formation in Cholesteric Shells G. Negro1 L.N. Carenza2 G. Gonnella3 D. Marenduzzo1 and E. Orlandini4 1SUPA School of Physics and Astronomy University of Edinburgh.pdf

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Topological Phases and Curvature-Driven Pattern Formation in Cholesteric Shells G. Negro1 L.N. Carenza2 G. Gonnella3 D. Marenduzzo1 and E. Orlandini4 1SUPA School of Physics and Astronomy University of Edinburgh

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