SUPPLEMENTARY INFORMATIONdoi: 10.1038/nmat2592

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SUPPLEMENTARYMATERIAL

Three-dimensional structure and multistable optical

switching of triple twisted particle-like excitations

in anisotropic fluids

1,2,3,*Ivan I. Smalyukh,

2,4Yves Lansac,

1,2Noel A. Clark, and

1,2Rahul P. Trivedi

1.Freeenergyanddirectorstructure

Weusenumericalminimizationoffreeenergytoobtainthestaticequilibriumandmetastable

configurations of )(ˆ rn

in confinedCNLCs. The spatio‐temporal evolution of the T3 structures

being generated by LG beams and during the unwinding transition under the action of an

externalelectricfieldhasbeenexploredusingcomputersimulationstooandwillbediscussed

indetailselsewhere.The freeenergydensity foraCNLCofpitchpunderanexternalelectric

field

€

E

LGoftheLGbeamisgivenby

€

ftotal = felastic + ffield ,where

€

felastic =K

11

2(∇ ⋅ ˆ n )

2+

K22

2[ ˆ n ⋅ (∇ × ˆ n ) ±

2π

p]

2+

K33

2[ ˆ n × (∇ × ˆ n )]

2−K

24[∇ ⋅ [ ˆ n (∇ ⋅ ˆ n ) + ˆ n × (∇ × ˆ n )]]

€

f field = −ε

0Δε

2( E LG ⋅ ˆ n )

2

1Department of Physics, University of Colorado, Boulder, Colorado 80309, USA. *e-mail:

ivan.smalyukh@colorado.edu

2Liquid Crystal Materials Research Center, University of Colorado, Boulder, Colorado 80309,

USA. 3Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309,

USA. 4Laboratoire d’Electrodynamique des Matériaux Avancés, Université François Rabelais-CNRS-

CEA, UMR 6157, 37200 Tours, France 

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The

€

K24‐termdescribesthesaddle‐splayelasticdeformationsandisknowntoplayakeyrolein

stabilizingthebluephases(BPs)formedbya3‐dimensionalcrystallineorganizationofdouble‐

twistcylinders.13Thesaddle‐splayconstant

€

K24 isdifficult tomeasureexperimentallybut it is

reasonabletoassumethat

€

K24

= K22.49,50

22

oenn −=Δε isthedielectricanisotropyoftheCNLCat

theusedlaserfrequencysuchthat )(ˆ rn

alignsparallel(

€

Δε > 0)to

€

E

LG .

Theminimizationofthefreeenergytofindtheequilibriumdirectorfield isperformed

usingarelaxationmethod.29Equilibrium3D‐structuresof )(ˆ rn

have

€

δF /δni= 0,where

€

niisthe

projectionofthedirector )(ˆ rn

ontothe

€

i ‐axis(

€

i=1(x),2(y),3(z))and

€

δF /δni arethefunctional

derivatives of the Frank free energy defined as

€

F = ftotal∫ dV withV being the volume of the

sample.Fromanumericalpointofview, thespatialderivativesof thedirectorarecomputed

usinga4thorder finitedifferencescheme.Periodicboundaryconditionsareappliedalong

€

ˆ x ‐

and

€

ˆ y ‐directionswhile fixedhomeotropicboundaryconditionareusedalong

€

ˆ z ‐direction.At

each stepΔt, the functional derivatives

€

δF δni are computed and the resulting elementary

displacement

€

δni definedas

€

δni= −Δt

δF

δni

isprojectedontothesurfacesn2 = 1andistakeninto

accountonlyifitleadstoadecreaseintheFrankfreeenergy

€

F .Otherwise,theincrement

€

Δt is

decreased.Thisprocedureisrepeateduntiltheelementarydisplacementissmallerthanagiven

valuesetto10‐8.Thediscretisationisdoneonfairlylargegrids(NxxNyxNz)withNx=Ny=119

andNz=35,orNx=Ny=239andNz=71inordertomakesurethattheminimum‐energy )(ˆ rn

isindeedastructurelocalizedinspaceinequilibriumwiththesurroundinguntwistedCNLCand

thattheperiodicboundaryconditionsdonotintroduceartifactsaffectingitsstability.Usinggrid

spacing such as hx = hy = hz = 0.05µmandNz = 71 gives sample thickness d = 3.50µm.All

presented simulations have been done for material parameters of nematic host ZLI‐3412

providedintheTable1.

Two different types of initial conditions were tested as a starting point of the

minimizationproceduredescribedabove,bothleadingtotheequilibriumToronstructures.The

firsttypecorrespondstoadirectorfieldconfigurationclosetotheonededucedfromtheFCPM

experiments(Figs.1,3,andSupplementaryFigs.S1‐S3).Thesecondtypeisobtainedbyusing

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electricfieldsofLaguerre‐Gaussianbeamsthathavetorus‐likeintensitydistributions(Fig.1g‐j)

and that induce toroid‐shaped initial deformations of the director field depending on the

charge l andradiusRofthevortexbeaminitslateralplane.Inagreementwithexperiments,

the initial location of the beam’s focal plane across the sample’s thickness does not have a

significant effect on the spatial location of the generated Toron. Although most of the

numericalresultsthatwepresentinthisworkhavebeenobtainedfor 1/ =pd ,wewereableto

stabilize T3‐like structures for a range of values

€

d / p = 0.75 −1.3, and find them being

minimum‐energy structures for d/p=0.9‐1.3 (Supplementary Fig. S4a). The Torons are well‐

definedlocalizedstructuresfor

€

d / p ≤1withtheirreducedlateralsize

€

L / p comparableto

€

d / p

(SupplementaryFig.S4b),where

€

L isthediameterofthetwist‐escapedλ‐disclinationring.For

€

d / p >1, the Torons have an extended disclination ring and their reduced lateral size >d/p

(Supplementary Fig. S4b). The relative energy of a 2D‐hexagonal lattice of T3‐1s, 2D linear

arraysofcholestericfingersofthefirstandsecondkinds(CF1andCF2,respectively),18,19

aswell

astranslationally invariantcholesteric (TIC)anduntwistednematic (Nem)configurations18are

comparedinFig.S4a(ascomputedforthevolumeofaunitcellofthehexagonallatticeformed

byToronsofthelargestobserved

€

L / p).TheseresultsshowthatT3‐1canbestabilizedwithina

broadrangeofd/p=0.75‐1.3,correspondingtoeithertheglobalorlocalfreeenergyminimum.

2.TopologicalSkeleton.

Thecriticalpointsand the topological skeletonarecomputedby formerly rewriting the

directorfieldasasetofordinarydifferentialequations(ODEs):

€

d x dt = ˆ n (x, y,z),with

€

t being

anarbitraryparameter(notthetime).Thisapproachconnectsthedynamicalsystemtheoryand

the director field and allows applying qualitative theory of differential equations in the

“physical space” rather than the phase plane of solutions of a system of ODEs. Similar

approaches are extensively used, for example, in the field of computational fluid

dynamics.31,32,51,52

Stationary points

€

x c (xc,yc,zc ) are such that

€

ˆ n ( x

c) = 0 . We can further

investigate the structure of the trajectories close to the stationary point by examining the

Jacobianmatrixofthepartialderivativesofthedirectorfield,

€

J ˆ n = ∂ni ∂x j with

€

(x j ≡ x,y,z).If

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thismatrixisnon‐singular,thestationarypoint

€

x c (xc,yc,zc ) isacriticalpoint.

51Theeigenvalues

andeigenvectorsoftheJacobiandescribethe localbehavioraroundacriticalpoint.Wehave

usedthenumericallibrarypfskel53tofindthepositionsofthecriticalpointsC1andC2,located

nearthetopandthebottomglassplates,respectively.BothC1andC2haveonerealeigenvalue

andtwocomplexconjugateeigenvalues.Thetwoeigenvectorscorrespondingto thecomplex

eigenvaluesdefineaplaneroughlyparalleltotheglassplateswhilethedirectionassociatedto

therealeigenvalue is roughlyalongthe

€

ˆ z ‐axis.Thecriticalpointsarehyperbolicandthe fact

thatwehavetwocomplexconjugateeigenvaluesindicatesthatthedirectorfieldspiralsaround

these defects.51We have computed the streamlets (flow lines representing themotion of a

massless particle) tangent to the director field. The streamlets originate very close from the

criticalpoint(defect) locationsandmovealongtheeigendirectionsofC1andC2.Equationsof

motionaresolvedusingeithera2ndora4

thorderRunge‐Kuttaforwardorbackward(depending

whether we are moving along a repulsive or an attractive direction) integrator. These

calculationshavebeenperformedusingpfskelandOpenDX(theopensourceversionofIBM’s

DataExplorerhasbeenusedforvisualizationspresentedinthiswork).Thetwoisolatedcritical

points with the streamlets as well as the twist‐escaped disclination ring (axis of the torus)

definethetopologicalskeletonoftheToronstructureshowninFig.4f.

The detailed description of computer simulations (using both vectorial and tensorial

approachesforminimizationofelasticfreeenergy54‐56

)comparedtotheexperimentalstudyof

directorfieldconfigurationsaswellastopologicalanalysisfortheexperimentally‐observedT3‐

2sandT3‐3swillbereportedelsewhere.

SUPPLEMENTARYREFERENCES

Numberingcontinuesfromthereferencelistinthemaintextofthemanuscript.

49.Allender,D.W.,Crawford,G.P.,andDoane,J.W.Determinationoftheliquid‐crystalsurface

elasticconstantK24.Phys. Rev. Lett.67,1442‐1445(1991).

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50.Polak,R.D.,Crawford,G.P.,Costival,B.C.,Doane,J.W.andZumer,S.Opticaldetermination

ofthesaddle‐splayelasticconstantK24innematicliquidcrystals.Phys. Rev. E49,R978(1994).

51. Asimov, D. Notes on the topology of vector fields and flows. Tech.  Rep.,  NASA  Ames 

Research Center.RNR‐93‐003(1993).

52.Globus,A.,Levit,C.andLasinski,T.Atoolforvisualizingthetopologyofthree‐dimensional

vectorfields.Proc. IEEE Visualization ’91, IEEE Computer Society Press,33‐40(1991).

53. Cornea, N.D., Silver, D., Yuan, X. and Balasubramanian, R. Computing hierarchical curve‐

skeletonsof3Dobjects.The Visual Computer21,945‐955(2005).

54.Anderson,J.E.,Watson,P.E.&Bos,P.J.LC3D:Liquidcrystaldisplay3‐Ddirectorsimulator

softwareandtechnologyguide(ArtechHouse,Boston,2001)

55.Gil., L. J.Numerical resolutionof the cholestericunwinding transitionproblem. J.  Phys.  II 

France5,1819‐1833(1995)

56.Sonnet,A.Kilian,A.&Hess,S.Alignmenttensorversusdirector:Descriptionofdefects in

nematicliquidcrystals.Phys. Rev. E52,718‐722(1995)

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SUPPLEMENTARYFIGURES

FigureS1.ComputersimulationsandFCPMimagingofthein‐planecross‐sectionsoftheT3‐1

structure.a,Schematicrepresentationofthecellwiththehyperbolicpointdefectsshownby

bluedotsandthetwistescapednon‐singulardisclinationringshownbyaredline.b,Computer

simulated )(ˆ rn

inthecentralplane(b‐bcross‐sectionshownina)oftheT3‐1structurecoplanar

withthedisclinationring.c,f,In‐planecross‐sectionsof )(ˆ rn

inthevicinityofpointdefectsnear

thebottomplate,c,andthetopplate,f.d,g,Correspondingsimulatedande,h,experimental

FCPM textures. The red bars ineand h indicate the corresponding FCPM linear polarization

states.

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FigureS2.FCPM imagingof theT3‐2swithdifferent locationsanddiametersofdisclination

rings. a‐c, FCPM vertical cross‐sections of three different T3‐2s in a cell of thickness

€

d =15µmwithdifferentdiametersofthedisclinationringatthebottomsurface.d,e,3DFCPM

imagesof the T3‐2 structures having thedisclination ringsd at thebottomande at the top

surface.

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FigureS3.Computer‐simulatedverticalcross‐sectionsoftheT3‐1structure.a,Thestructure

consistsoftheradialtwistof inthecentralplaneofthecellaswellastwopointdefects

closetothesubstrates;theinsetsshowsimulated inmutually‐orthogonalxzandyzcross‐

sections intersecting the hyperbolic point defects at the top and bottom glass plates. b,

Computer‐simulateddirectorfieldintheaxialcross‐section.

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Figure S4. Elastic free energy and the reduced diameter of the T3‐1 configuration as a

functionofd/p.a,FreeenergyofahexagonalarrayofT3‐1scomputedforaunitcellvolumeof

€

≈ 38µm3 as compared to that of an equivalent sample volume with the twisted invariant

configuration (TIC)18,19

, untwisted nematic‐like structure (Nem) and linear arrays of two

differentcholestericfingersCF1andCF218,19

(thedensityiscalculatedforthevolumeoftheunit

cellofa2Dhexagonal lattice formedbyTorons).b,Normalizedequilibriumdiameter

€

L / p of

theT3‐1structureasafunctionof pd / .