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Liquid Crystals

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Computing equilibrium states of cholestericliquid crystals in elliptical channels with deflationalgorithms

David B. Emerson, Patrick E. Farrell, James H. Adler, Scott P. MacLachlan &Timothy J. Atherton

To cite this article: David B. Emerson, Patrick E. Farrell, James H. Adler, Scott P.MacLachlan & Timothy J. Atherton (2018) Computing equilibrium states of cholesteric liquidcrystals in elliptical channels with deflation algorithms, Liquid Crystals, 45:3, 341-350, DOI:10.1080/02678292.2017.1365385

To link to this article: https://doi.org/10.1080/02678292.2017.1365385

© 2017 The Author(s). Published by InformaUK Limited, trading as Taylor & FrancisGroup.

Published online: 22 Aug 2017.

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Computing equilibrium states of cholesteric liquid crystals in elliptical channelswith deflation algorithmsDavid B. Emersona, Patrick E. Farrellb, James H. Adlera, Scott P. MacLachlanc and Timothy J. Athertond

aDepartment of Mathematics, Tufts University, Medford, MA, USA; bMathematical Institute, University of Oxford, Oxford, UK; cDepartment ofMathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada; dDepartment of Physics and Astronomy, TuftsUniversity, Medford, MA, USA

ABSTRACTWe study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system isgeometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation,but the elliptical domain precludes this. The frustration is resolved by deformation of the layers orintroduction of defects, leading to a particularly rich family of equilibrium configurations. To identifythe solution set, we adapt and apply a new family of algorithms, known as deflation methods thatiteratively modify the free energy extremisation problem by removing previously known solutions. Asecond algorithm, deflated continuation, is used to track solution branches as a function of the aspectratio of the ellipse and preferred pitch of the cholesteric.

ARTICLE HISTORYReceived 14 June 2017Accepted 6 August 2017

KEYWORDSCholesterics; confinement;defects; simulationtechniques

1. Introduction

Cholesteric liquid crystals are complex fluids that exhi-bit long-range orientational order, elasticity, localalignment at surfaces, optical activity and response toexternal stimuli [1]. They are composed of chiral mole-cules that, in the absence of boundaries, adopt a helicalstructure with a preferred pitch, q0, set by the molecu-lar structure and the ambient temperature. There hasrecently been a great deal of interest in cholesterics inconfined geometries because of parallels with othercondensed matter systems such as chiral ferromagnets,Bose–Einstein condensates and the Quantum Halleffect. All of these systems exhibit topological defectsunder confinement, including skyrmions and torons,where the boundary conditions preclude adoption ofthe energetically preferred uniformly twisted state.

Hence, they are geometrically frustrated. It was recog-nised some time ago that nematic liquid crystals alsomay potentially form skyrmions, but these are onlymetastable due to the lack of preferred twist [2].

Liquid crystals are particularly attractive to studythese defect structures, because they can be conveni-ently produced and imaged in three dimensions withtechniques such as confocal microscopy [3].Cholesterics may form skyrmion lattices in two dimen-sions [4]. In three dimensions, torons resemble parti-culate inclusions [5,6] and form chains or lattices [7].Other more complicated defect structures called ‘twis-tions’ occur in films thinner than the cholesteric pitch[8]. They also provide an elegant experimental actuali-sation of the Hopf fibration, a map from the 3-sphereonto the 2-sphere such that each distinct point of the 2-

CONTACT Timothy J. Atherton timothy.atherton@tufts.edu

LIQUID CRYSTALS, 2018VOL. 45, NO. 3, 341–350https://doi.org/10.1080/02678292.2017.1365385

© 2017 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properly cited.

sphere comes from a distinct circle of the 3-sphere [9],and are an ideal model system for experimentalexploration of Hopf solitons [10]. Strongly confinedgeometries such as micropatterned surfaces [11], chan-nels [12,13] and droplets [14] can all be used to controland order the location of skyrmions.

A key challenge in simulating these systems is that,due to the geometric frustration, they possess a parti-cularly rich family of stationary solutions of the freeenergy. The ground state strongly depends on theshape of the domain and material parameters, includ-ing the cholesteric pitch. Typically, extremisation isperformed from an initial guess using a relaxationalgorithm, or by solving a set of non-linear Euler–Lagrange equations. In either case, having found asolution, the question remains: are there others? It isalso highly desirable to track the solution set as afunction of geometric and material parameters toassemble a bifurcation diagram.

A common approach to identify distinct solutions,referred to in the mathematics literature under theumbrella term of multistart methods [15], is to beginfrom a significant number of initial guesses. Thisrequires extensive knowledge of the problem todevise a suitable set of initial guesses and can beinefficient as multiple guesses often converge to thesame configuration. Other well-established techniquesinclude numerical continuation [16,17], which is par-ticularly effective in fully resolving connected bifur-cation branches but can neglect solutions if they arenot homotopic with respect to the continuation para-meters [18], and approaches, such as Branin’smethod, that rely on numerical integration of theDavidenko differential equation corresponding tothe original non-linear problem [19,20]. Branin’smethod is capable of systematic computation of mul-tiple solutions but requires determinant calculationsthat become prohibitively expensive for large-scaleproblems without special structure.

In this paper, we adapt a new technique known as thedeflation method [21,22] to a model problem in thisclass, the configuration of a cholesteric in an ellipticalchannel. The method is generalisable, robust and com-putationally efficient for large-scale applications. It hasbeen successfully applied to a diverse set of non-linearproblems including non-linear partial differential equa-tions (PDEs), singularly perturbed problems, the analy-sis of Bose–Einstein condensates and the computation ofdisconnected bifurcation diagrams [18,21,23,24]. Thispaper is structured as follows: in Section 2, we brieflydescribe the problem; in Section 3, we introduce thedeflation technique with an illustrative toy example. InSection 4, we present results for the cholesteric problem.

Conclusions are drawn, with prospects for furtherenhancement of the algorithm, in Section 5.

2. Model

We consider a cholesteric liquid crystal in a channelwith elliptical cross section. Equilibrium structures arefound by identifying critical points of the Frank energy,

F ¼ 12�ΩdV K1 � � nð Þ2 þ K2 n � �� nþ q0ð Þ2

þ K3 n� �� nj j2; (1)

where K1, K2 and K3 are the splay, twist and bend elasticconstants; n is a headless unit vector, the director, thatcorresponds to the local symmetry axis of the molecularorientational distribution and q0 is the preferred pitch forthe cholesteric. Rigid anchoring (Dirichlet) boundaryconditions are imposed on the boundary @Ω, where thedirector is required to point perpendicular to the plane ofthe cross section. The energy is readily non-dimensiona-lised by introducing a typical length λ and dividingthrough by a characteristic magnitude of the elastic con-stants ~K; henceforth, we use dimensionless parameters.

As discussed in the introduction, this problem pro-motes the existence of multiple local equilibria by con-struction. To see why, first consider the cholesteric in theabsence of boundaries. As is well known, the minimiser of(1) is a unique uniformly twisted state. Level sets of n formfamilies of equally spaced planes often referred to as cho-lesteric ‘layers’. We note a valuable discussion of the lim-itations of this view is found in Ref. [25]. Variation awayfrom this preferred structure, which is equivalent to com-pressing or bending the layers, implies an elastic cost.Considering a cholesteric in a disc, the minimisers of (1)are solutions where n rotates about the radial axis and lieseverywhere perpendicular to it. The number of rotations isdetermined by the cholesteric pitch, q0, which promotes aconstant rotation rate, and level sets of constant orientationtherefore form equally spaced concentric circles.

For an elliptical domain, however, it is not possible tofill the ellipse with equally spaced layers, and so defectsor a variable layer spacing must be introduced. Thecholesteric order, which prefers a uniformly twistedstate, and the shape of the domain are in competition,so the system is said to be geometrically frustrated. Thefrustration is resolved by adopting a compromise state,incorporating some combination of layer deformationor defects; in common with other frustrated systems,there is typically more than one way to do this, leadingto the possibility of more than one minimiser.

Further, we explore solutions where K2 > K1;K3,which might occur in exotic liquid crystal systems

342 D. B. EMERSON ET AL.

[26]. This choice of parameters leads to a material thatis doubly frustrated because it is required to twist by thecholesteric term but, nonetheless, the twist is relativelyexpensive compared to other deformation modes. As aresult, the cost of modulating the cholesteric layers isreduced. The interaction of geometric and internalfrustration is expected to lead to a particularly richsolution set because they permit multiple ways ofrelieving the frustration: one solution might accommo-date an incommensurate number of cholesteric periodsby folding the layers; another might introduce a defect.These parameters therefore yield an extremisation pro-blem that we anticipate a priori to be very challengingto explore by naive multistart methods.

3. Deflation

In solving problems that possess multiple equilibriumsolutions, such as that posed in Section 2, a key challengeis to ensure that the true ground state has been found fromthe set of energetically low-lying solutions. The idea of thedeflation algorithm is to successivelymodify the non-linearproblem under consideration to eliminate known solu-tions, enabling the discovery of additional distinct solu-tions. Consider AðuÞ ¼ 0, a set of non-linear equations,that may admit multiple solutions. This system, forinstance, could represent a set of continuous or discretisednon-linear PDEs; here, we minimise the Frank energy in(1), subject to the unit-length constraint of the director,and consider the resulting system of non-linear first-orderoptimality conditions. Using a known solution, v, to thesystem AðvÞ ¼ 0, we define the deflation operator,

Mp;αðu; vÞ ¼ 1

u� vk kpUþ α

!I; (2)

where α � 0 is a shift scalar, p 2 ½1;1Þ is the defla-tion exponent and I is the identity operator. An appro-priate norm �k kU must be chosen for the solution spaceto which the solutions belong. The deflated system isthen formed by applying the deflation operator to theoriginal non-linear system as

BðuÞ ¼ Mp;αðu; vÞAðuÞ ¼ 0: (3)

Iterative techniques, such as Newton’s method, maythen be applied to solve the deflated system. Whilethese iterations are guaranteed to not converge to theknown solution v under mild regularity conditions, theremainder of the solution space for the original systemis preserved by the deflation operator. Numericalexperiments have found the effectiveness of the defla-tion operator to be relatively insensitive to the choiceof deflation parameters. However, for certain problems,

performance improvements may be obtained by vary-ing p and α [21,22]. Typical values, used throughoutthis paper, are α ¼ 1 and p ¼ 2.

Having found two solutions v1; v2, an expandeddeflation operator can be constructed by compositionof single-solution deflation operators,

Mp;α u; v1; v2ð Þ ¼ Mp;α u; v1ð ÞMp;α u; v2ð Þ; (4)

and applied to the original non-linear system. As the setof known solutions v1; v2; . . . ; vmf g is expanded, thedeflation operator is grown as the product of the singledeflation operators for each distinct solution in the set,

Mp;α u; v1; v2; . . . ; vmð Þ ¼Ymi¼1

Mp;α u; við Þ; (5)

and its action remains multiplicative on the originalsystem.

To provide a simple and tractable illustration of thedeflation process, we apply it to the problem of locatingcritical values of a one dimensional objective function,

f ðxÞ ¼ 154

x4 � 152

x2 þ 1; (6)

displayed in Figure 1 by solving the equation,

f 0ðxÞ ¼ 454

x3 � 252

x ¼ 0: (7)

Starting from the initial guess x ¼ 2:2, Newton’s methodlocates the first solution x0 ¼

ffiffiffi5

p=2. The deflation opera-

tor is constructed following the definition in (2) as

Figure 1. Toy 1D example of deflation. Critical values of thefunction fðxÞ (black), found by solving f 0ðxÞ ¼ 0 (grey). Havinglocated a minimum at x0 ¼

ffiffiffi5

p=2, a deflated function f 0dðxÞ

(grey; dashed) is constructed; this now contains a pole at x0but retains zeros in common with f 0ðxÞ.

LIQUID CRYSTALS 343

Mp;α x; x0ð Þ ¼ 1

x � x0j jp þ α; (8)

where �j j denotes the standard absolute value. Applyingthis to (7) yields the deflated optimality condition,

0 ¼ f 0dðxÞ ¼ Mp;α x; x0ð Þf 0ðxÞ¼ 1

x � x0j jp þ α

� �454

x3 � 252

x

� �; (9)

to be solved for x. The function f 0dðxÞ is also plotted inFigure 1 for deflation parameters α ¼ 1 and p ¼ 2.Notice that x0 is not a solution to f 0dðxÞ ¼ 0, while theremaining solutions to f 0ðxÞ ¼ 0 persist as solutions tof 0dðxÞ ¼ 0. Use of the deflation operator precludes con-vergence of certain iterative techniques, such asNewton’s method, to x0 while facilitating convergenceto additional distinct solutions from the same initialguess. With the deflation parameters chosen previouslyand the same initial point, x ¼ 2:2, Newton’s methodconverges to the critical point x1 ¼ 0:0. Thus, twosolutions are obtained from the same initial guess.The process may then be repeated by constructing amulti-deflation operator, incorporating both knownroots, to enable the discovery of the third distinctsolution at x2 ¼ � ffiffiffi

5p

=2 to (7) and hence identify allextremal values of f ðxÞ.

While deflation is a useful device for findingdistinct solutions, the number of solutions discov-ered may still depend on the analyst supplying asuitable set of initial guesses. A systematic way togenerate the set of initial guesses to use is providedby continuation. Suppose that the problem incorpo-rates some set of parameters k, which for the liquidcrystal problem includes the elastic constants andpreferred pitch q0. Given a set of solutions for someinitial value of these parameters k0 ¼ �k0, we useeach of these solutions as an initial guess forNewton’s method at a nearby parameterk0 ¼ �k0 þ δk, deflating each solution as we find it.Subsequently, we use the full power of the deflationapproach to search for new solution branches that,if discovered, can be extended to other values of k0using standard continuation techniques. The solu-tions at k0 ¼ �k0 þ δk can in turn be used as initialguesses to find the solutions at k0 ¼ �k0 þ 2δk, andso on. This combination of deflation and continua-tion is referred to as deflated continuation and is aneven more powerful algorithm than standard defla-tion applied to a single non-linear problem [18]. Itcan be interpreted physically as computing a bifur-cation diagram, a portrait of the solutions of anequation as a parameter varies. We shall use both

the deflation and the deflated continuationapproaches to resolve ground state solutions of thecholesteric problem in the next section.

4. Results

We apply the deflation algorithm described above tothe cholesteric problem in an ellipse, varying the aspectratio of the domain, μ, and preferred cholesteric pitch,q0. In each simulation, the director is held fixed on theboundary such that the director points out of the plane,i.e. n ¼ ð0; 0; 1Þ. Elastic constants are chosen to beK1 ¼ 1, K2 ¼ 3:2 and K3 ¼ 1:1, corresponding to theexotic splay–bend cholesteric described above. Thecomputational domain is centred on the origin withmajor axis parallel to the x-axis. The area of eachbounding ellipse is held fixed at 3π=2.

Our multilevel finite-element code used to com-pute stationary points of the free energy (1) is thor-oughly described elsewhere [22,27–29]. Briefly, thecode uses the Cartesian representation of the directorn ¼ nx; ny; nz

� �and directly finds equilibrium points

of the Frank energy (1) by applying Newton linear-isation to the first-order optimality conditions invariational form, resulting from the constrainedminimisation. The code is based on deal.II [30] andfeatures mesh refinement and nested iteration [31]such that the problem is discretised and solved firston coarse meshes where computation is cheap, resol-ving major solution features, and then interpolatedto more refined meshes to determine finer attributesof the computed approximation. Nested iteration hasbeen shown to significantly improve computationalefficiency in a wide variety of problems includingliquid crystal simulations [29,32,33]. Here, we use anested iteration mesh hierarchy of four mesh levelsof refinement with 4884 degrees of freedom on thecoarsest grid and ending with 297,988 degrees offreedom at the finest level. Finally, a damping factor,ω, is applied to each Newton step for both theundeflated and deflated systems. This dampedNewton stepping is combined with an increasedstep size, �ω, when the non-linear residual dropsbelow 0.1. The accelerated Newton stepping isapplied to increase the rate of convergence whenthe candidate solution begins to closely satisfy theoptimality conditions.

As a first example, in Figure 2, we display the resultsof a typical run for aspect ratio μ ¼ 1:5 and q0 ¼ 8.The algorithm is initialised from three initial guesses(shown in the left column of Figure 2). As anticipated,deflation enables the discovery of several solutions for

344 D. B. EMERSON ET AL.

each value by successively removing them with thedeflation operator. Several of these solutions possessenergetically degenerate partners that are obtained bysimple reflection about an axis of the ellipse. These arealso found by deflation, even though knowledge of thesymmetry of the problem is not explicitly built into thealgorithm.

It is important to note that the solutions to whichthe code converges are stationary points of theLagrangian (Frank energy plus unit-length constraint),not necessarily minimisers of the energy. It is thereforehighly desirable to characterise the nature of each solu-tion as it is found, i.e. determine if it is a local mini-mum, a local maximum or a saddle point. To do this,we must verify the second-order sufficiency conditions:a stationary point is a local minimum if the reducedHessian of the energy (the Hessian projected onto thenullspace of the linearised constraints, i.e. restricted tofeasible perturbations) is positive definite. Oneapproach would be to assemble the linearised con-straint Jacobian, compute a (dense) basis for its null-space using the singular value decomposition,construct the (dense) reduced Hessian and computeits eigenvalues; but this would be unaffordably expen-sive. A better way is to exploit the fact that the eigen-values of the reduced Hessian of the energy are relatedto the eigenvalues of the (sparse) Hessian of theLagrangian: by counting the number of negative eigen-values of the Hessian of the Lagrangian, and comparingit to the dimension of the constraint space, we candetermine the inertia (the number of positive, zeroand negative eigenvalues) of the associated reducedHessian of the energy [34, Thm. 16.3]. This allows forthe characterisation of the nature of each solutionfound using a single sparse LDLT decomposition,

computed using the FEniCS, PETSc and MUMPS soft-ware packages [35–37].

In Figure 3, we show the computed ground state(lowest energy solution) as a function of μ and q0. Foreach solution, we display the value of the energy func-tional and also compute the Skyrmion number [2],

Q ¼ 14π

� n � @n@x

� @n@y

� �dA; (10)

a topological index that represents the number of timesn covers the unit sphere. Such indices help identifytopologically distinct solutions: as the parameters μand q0 are slowly varied in Figure 3, the ground statemostly changes smoothly. However, between certainvalues, e.g. q0 ¼ 4 and 5 with μ ¼ 1, a transition to anew solution as the ground state occurs; this is accom-panied by a change in the Skyrmion number. Somesolutions that are quite different in appearance, e.g.μ ¼ 1:85, between q0 ¼ 4 and 5 or q0 ¼ 8 and 9 haveidentical Q because they are linked by a continuousdeformation of the director field. While deflated con-tinuation will enable us to find intermediate solutionsbetween chosen values, and resolve any transitions thattake place, the Skyrmion number provides a classifica-tion of the distinct branches that arise.

For certain values of μ and q0, the solution setdiscovered by deflation alone failed to include a mini-mal energy solution that was stable. These values areindicated in Figure 2 with an asterisk. For these values,we applied the deflated continuation techniquedescribed above to identify the stable ground stateshown in Figure 3. For instance, consider the case μ ¼1:15 and q0 ¼ 7. The lowest energy solution foundusing deflation is displayed as an inset indicated byan asterisk in Figure 4 but possesses unstable directions

10.506

8.730

10.872 11.118

9.067 18.742 (a)

(b)

(c)

Figure 2. (Colour online) Example solution set found with deflation. Solution set for cholesteric pitch q0 ¼ 8 and aspect ratioμ ¼ 1:5. Rows A–C depict different initial guesses (left) and the solution set (right) recovered for each through successiveapplications of the deflation operator (5). The computed free energy of each solution is also given.

LIQUID CRYSTALS 345

according to the Hessian analysis described above. Wetherefore use the μ ¼ 1:15 and q0 ¼ 6 solution set andcontinue in q0 from 6 to 7. The energies of intermedi-ate solutions obtained in the process are plotted inFigure 4 as a bifurcation diagram; the initial and finalsolution set recovered in this process are displayed inFigure 4 as insets. A new pair of solutions, not in theq0 ¼ 6 solution set, emerges through a fold bifurcationat approximately q0 � 6:67 and represents the stableground state at higher values of q0; the prior groundstate becomes higher in energy and is now unstable.

The same procedure was applied to all the proble-matic cases in Figure 3, where the solutions shown arethe lowest energy solutions found and are all verified asstable. This example illustrates the power of deflatedcontinuation to track distinct branches in the solutionset and identify solutions very different from the initialguesses provided. While it remains possible that thetrue ground state remains elusive for some values inμ; q0ð Þ space, it is clear that deflation and deflatedcontinuation are powerful tools to assist in the assem-bly of phase diagrams.

The solutions found in Figure 3 catalogue theinteresting interplay of the elastic constants, choles-teric parameter and confining geometry. For μ ¼ 1,a circular domain, increasing q0 initially leads only

to the incorporation of additional rotations asexpected. Around a critical value of q0 ¼ 7, thecontours of constant orientation are greatlydeformed as the number of radial rotations in the

q=3 4 5 6 7

µ=1

1.15

1.35

1.5

1.65

1.85

2.0

2.123 5.833 5.316 2.941

5.476

7.633

5.809 2.927 6.246

8

7.536 3.533 5.004 7.508 4.309 5.496

9.231 4.714 4.068 8.32

3.976 3.384 7.482 6.413 4.338

5.779

7.033 3.342 7.488 11.216

3.194 7.626 9.138

7.813 9.348 13.001

0 1 1 0 1 0 1 2

9

5.054 6.079 11.079 13.423

1 1 1 0 0 0 0 1

10

6.588 5.433 10.535 11.761

1 1 1 0 0 0 0 1

6.349 4.631

3.717 5.890 8.602

1 1 0 0 0 1 1 1

6.605 2.255

2.798

3.466 3.822

0 0 0 0 1 11 1

4.482

0 0 0 1 1 1 1 1

4.500 3.972

1 0 0 0 1 1 1 1

* *

* *

* *

*

* *

*

Figure 3. (Colour online) Cholesteric liquid crystal in an elliptical domain. Ground state solutions shown as a function of pitch q0and aspect ratio μ. For each solution, the value of the energy functional is displayed in the top right, with the Skyrmion numbershown bottom right. Solutions indicated by *, marked top left, were found using the deflated continuation technique.

Cholesteric pitch q0

Ene

rgy

func

tiona

l

*

Figure 4. (Colour online) Bifurcation diagram as a function ofq0 generated by deflated continuation for aspect ratioμ ¼ 1:15. The solution set is visualised at q0 ¼ 6 and q0 ¼ 7.Black points represent stable solutions and grey points indicateone unstable direction. The lowest energy, yet non-stable,solution identified by deflation without continuation for q0 ¼7 is indicated by an asterisk.

346 D. B. EMERSON ET AL.

channel increases from π at q0 ¼ 6 to 3π=2 forq0 ¼ 8. A similarly deformed structure is visible atq0 ¼ 10, which is apparently close to a jump from3π=2 rotations to 2π.

For higher aspect ratios, the contours of constantorientation can be deformed by the geometry of thechannel. For example, for aspect ratio μ ¼ 1:5 andq0 � 7, the ground state consists of the directorrotating by 2π about the radial direction from thecentre; above this value, an extra π rotation isincorporated. Comparing the shape of contours ofconstant tilt, notice that the interior ‘layer’ isapproximately circular for q0 ¼ 5 but becomesmore elongated with increasing q0. For q0 ¼ 8, theground state is strikingly different: two highlydeformed interior circular layers are accommodatedwithin one contiguous outer layer. The ground statefor q0 ¼ 9 reverts to the expected pattern, simplyincorporating an additional twist. Furthermore, as μincreases, the transition points between states withdifferent amounts of twist occur at higher values ofq0 and are typically proceeded by substantial layerdeformation. Therefore, the confining geometryplays a role in deterring or encouraging the addi-tion of layers.

We note that deflation uncovers a particularlylarge number of solutions for q0 ¼ 8 and that manyof the solutions have relatively low energy comparedto the ground state. For other values of q0, only afew of the solutions are close to the ground state inenergy. We speculate that this phenomenon is due tocommensurability, with some values of q0 being morecompatible with the shape of the domain thanothers. For a circular domain, commensurate solu-tions exist where q0 happens to allow an integermultiple of π rotations from the centre. Maximallystrained solutions occur between these special values,potentially inducing deformation of the layers torelieve the frustration. This is clearly visible at μ ¼1:5 and q0 ¼ 8, or at μ ¼ 1:85 and q0 � 9, where theinner layer in both cases is highly tortuous to fill theinterior of the domain.

To resolve the sequence of transitions that occursaround one of the maximally strained solutions, wevisualise a bifurcation diagram in Figure 5 for anelliptical domain with aspect ratio μ ¼ 1:5 and withthe preferred pitch ranging from q0 ¼ 5 to q0 ¼ 9.We initialise the computation with the solutionsfound by our previous analysis. The diagram shownin Figure 5(a) displays a relatively small solution setfor q0 < 7, but above q0 � 7:4, a dense thicket ofadditional solutions appears. As is visible in the

expanded region depicted in Figure 5(b), this con-sists of a rapid succession of fold and pitchforkbifurcations. The corresponding solutions at severalvalues of q0 are displayed in Figure 5(c), illustratingthe striking complexity of solution sets that can beuncovered using deflated continuation.

Because deflation suppresses solutions close to theinitial guess, even very distant solutions can be recovered.A particularly important result is that while our initialguesses are smooth functions, the algorithm was able tospontaneously identify solutions with disclinations.Confined cholesterics in rectangular domains or channelshave been experimentally and numerically shown to exhi-bit structures with disclinations [5,12,13], as discussed inthe introduction. The existence, type and number of thedisclinations were shown to be modulated by changes tothe depth and width of the channel, as well as the choles-teric pitch. Figure 6 displays three examples for μ ¼ 1:5and varying q0 with computed director fields and asso-ciated elastic energy densities from which disclinationsare readily identified. For q0 ¼ 5, the solution shown inFigure 6(a) has four defect points, arranged in a diamondpattern near the centre of the domain. The solutions inFigure 6(b,c) were found for q0 ¼ 7 and possess two andone disclinations, respectively. Comparing the solutions’free energies, it is clear that the single defect structure isenergetically preferred. We note that the energy of thesolution in Figure 5(c) corresponds to the third lowestenergy (and first unstable state) shown for q0 ¼ 7 in thebifurcation diagram in Figure 5(a). Thus, our resultssuggest that the propensity of the cholesteric to formingmetastable structures with disclinations in elliptical chan-nels, perhaps upon quenching from the isotropic phase,strongly depends on the aspect ratio of the channelboundary. Moreover, our numerical experiments indicatethat multiple equilibrium configurations with distinctdisclination patterns may exist for the same geometryand material parameters.

The deflation technique plays a central role in thediscovery of these disclination arrangements. Forinstance, numerical simulations in Ref. [13] relied on apriori knowledge, gained from experimental observa-tions, that disclination structures should be present inorder to initialise the Newton iterations within a basin ofattraction. We emphasise that here the simulations areinitialised with smooth director fields. Multiple solutionsand the emergence of disclinations occurred as sponta-neous discoveries enabled by the deflation computa-tions. In situations where experimental and analyticalinformation is limited, such numerical capabilities facil-itate a more robust and thorough exploration of theadmissible solution space of a given problem.

LIQUID CRYSTALS 347

5. Conclusion

In this paper, we present a new technique, deflation, forrecovering equilibrium solutions of the free energy of aliquid crystal. The utility of the method is shown on a toyexample and then used to determine the structure of acholesteric in an elliptical domain. The ground state isidentified for a range of aspect ratios μ and preferredpitches q0, showing gradual deformation of the solutionsas a function of these parameters and transitions to

different solutions at critical values. For selected valuesof μ and q0, we reconstruct the bifurcation diagram forthe low energy solution set, finding remarkably densesolutions sets near the transition points. In future work,we will apply the method to characterise the solution setof more complex geometries involving cholesterics, suchas the rich 3D structures observed in Ref. [3]. It will alsobe important to include weak anchoring, and otherexperimentally relevant anchoring conditions such as

Cholesteric pitch q0

95 6 7 8

30

0

10

20

Ene

rgy

func

tiona

l(a)

7.4

5.88

5.90

7.34

7.55

6.62

6.76

6.87

7.59

7.75

7.46

7.79

7.81

7.85

8.15

8.58

7.85

7.79

8.10

8.23

8.34

8.42

8.48

8.91

8.84

8.03 8.22

8.32

8.53

8.87

9.18

9.29

9.40

9.93

8.77

8.74

8.82

8.88

8.97

9.38

9.76

10.04

10.07

Cholesteric pitch q0

10.30

10.39

7.4 7.6 7.8 8.0 8.2Cholesteric pitch q0

4

6

8

10

Ene

rgy

func

tiona

l

(b)

(c)

Figure 5. (Colour online) Deflated continuation. (a) Bifurcation diagram computed for fixed μ ¼ 1:5 and continuing in q0. Points arecoloured by the number of unstable directions, with black indicating a stable solution, and lighter grey indicating more unstabledirections. A particularly dense portion of the diagram, outlined in red, is shown in greater detail in (b) where vertical dashed linesindicate values of q0 for which the solution set is displayed in (c).

348 D. B. EMERSON ET AL.

degenerate tangential anchoring rather than the verticalDirichlet condition imposed here. The surface saddle-splay K24 elastic term must also be included togetherwith weak anchoring and will likely further modify thesolution set.

The deflation methodology significantly enhances theutility of Newton iterations applied to non-linear systemsby enabling Newton’s method to converge to multiplesolutions from the same initial guess. Applying the defla-tion operator ensures that subsequent Newton iterationsdo not converge to previously discovered solutions andeffectively modifies the basin of attraction to includeunknown solutions. While convergence to multiple solu-tions is not guaranteed through use of the deflation opera-tor, sufficient conditions for convergence of deflatediterations are constructed in Ref. [18] based on a general-isation of the Rall–Rheinboldt theorem. Our results notonly illuminate the effect of deflation but also highlight thecase that the use of multiple initial guesses to discover allsolutions is not entirely eliminated with deflation.However, deflated continuation systematically provides asequence of good initial guesses and increases overall effi-ciency and reliability of multiple solution discoverythroughmore systematic exploration of the solution space.

More generally, deflation and deflated continuationtherefore allows theorists to recover energetically low-lying solutions in which a liquid crystal may becomekinetically trapped. It can also be used to track differentsolution branches when a system exhibits a bifurcationwith respect to some external parameter, e.g. the appliedfield in a Freedericksz transition. The method is verygeneral and can be readily adapted to different represen-tations, e.g. the Q-tensor and parametrisations; it is likely

to be most useful in systems where little analytical gui-dance or experimental imaging is available.

Acknowledgements

The authors contributed to the work as follows: SPM devisedthe toy example in Section 3; DBE, PEF and JHA performedthe simulations in Section 4 and all authors contributed totheir analysis; TJA wrote the paper, which was revised by allauthors.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by NSERC discovery grant. TJA issupported by a Cottrell Award from the Research Corporationfor Science Advancement, and a CAREER award from theNational Science Foundation (award number DMR-CMMT-1654283). PEF is supported by an EPSRC Early Career ResearchFellowship (award number EP/K030930/1).

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(a) (b) (c)

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