SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION Three-dimensional structure and multistable optical switching of triple twisted particle-like excitations in anisotropic fluids 1,2,3,* Ivan I. Smalyukh, 2,4 Yves Lansac, 1,2 Noel A. Clark, and 1,2 Rahul P. Trivedi 1.Freeenergyanddirectorstructure Weusenumericalminimizationoffreeenergytoobtainthestaticequilibriumandmetastable configurations of nˆ ( r ) in confined CNLCs. The spatio temporal evolution of the T3 structures being generated by LG beams and during the unwinding transition under the action of an externalelectricfieldhasbeenexploredusingcomputersimulationstooandwillbediscussed in details elsewhere. The free energy density for a CNLC of pitch p under an external electric field E LG ofthelgbeamisgivenby f total = f elastic + f field,where f elastic = K 11 2 ( n ˆ )2 + K 22 2 [ n ˆ ( n ˆ ) ± 2π p ]2 + K 33 2 [ n ˆ ( n ˆ )]2 K 24 [ [ n ˆ ( n ˆ ) + n ˆ ( n ˆ )]] f field = ε 0Δε 2 ( E LG n ˆ ) 2 1 Department of Physics, University of Colorado, Boulder, Colorado 80309, USA. * ivan.smalyukh@colorado.edu 2 Liquid Crystal Materials Research Center, University of Colorado, Boulder, Colorado 80309, USA. 3 Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA. 4 Laboratoire d Electrodynamique des Matériaux Avancés, Université François Rabelais-CNRS- CEA, UMR 6157, Tours, France nature materials 1

2 The K 24 termdescribesthesaddle splayelasticdeformationsandisknowntoplayakeyrolein stabilizingthebluephases(bps)formedbya3 dimensionalcrystallineorganizationofdoubletwistcylinders. 13 Thesaddle splayconstant K 24 isdifficulttomeasureexperimentallybutitis reasonabletoassumethat K 24 = K , Δε = ne n o isthedielectricanisotropyofthecnlcat theusedlaserfrequencysuchthat nˆ ( r ) alignsparallel( Δε > 0)to E LG. Theminimizationofthefreeenergytofindtheequilibriumdirectorfieldisperformed usingarelaxationmethod. 29 Equilibrium3D structuresof nˆ ( r ) haveδf /δn i = 0,where n i isthe projectionofthedirector nˆ ( r ) ontothe i axis(i=1(x),2(y),3(z))andδf /δn i arethefunctional derivatives of the Frank free energy defined as F = f total dv with V being the volume of the sample. From a numerical point of view, the spatial derivatives of the director are computed usinga4 th orderfinitedifferencescheme.periodicboundaryconditionsareappliedalong ˆ x and ˆ y directionswhilefixedhomeotropicboundaryconditionareusedalong ˆ z direction.at each step Δt, the functional derivatives δf δn i are computed and the resulting elementary displacementδn i definedasδn i = Δt δf δn i isprojectedontothesurfacesn 2 = 1andistakeninto accountonlyifitleadstoadecreaseinthefrankfreeenergy F.Otherwise,theincrementΔt is decreased.thisprocedureisrepeateduntiltheelementarydisplacementissmallerthanagiven valuesetto10 8.Thediscretisationisdoneonfairlylargegrids(N x xn y xn z )withn x =N y =119 andn z =35,orN x =N y =239andN z =71inordertomakesurethattheminimum energy nˆ ( r ) isindeedastructurelocalizedinspaceinequilibriumwiththesurroundinguntwistedcnlcand thattheperiodicboundaryconditionsdonotintroduceartifactsaffectingitsstability.usinggrid spacing such as h x =h y =h z = 0.05 µm and N z = 71 gives sample thickness d = 3.50 µm. 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 2 nature MATERIALS

3 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 d / p = 1,wewereableto stabilize T3 like structures for a range of values d / p = , and find them being minimum energy structures for d/p= (Supplementary Fig. S4a). The Torons are welldefinedlocalizedstructuresfor d / p 1withtheirreducedlateralsize L / pcomparableto 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 astranslationallyinvariantcholesteric(tic)anduntwistednematic(nem)configurations 18 are comparedinfig.s4a(ascomputedforthevolumeofaunitcellofthehexagonallatticeformed bytoronsofthelargestobserved L / p).theseresultsshowthatt3 1canbestabilizedwithina broadrangeofd/p= ,correspondingtoeithertheglobalorlocalfreeenergyminimum. 2.TopologicalSkeleton. The critical points and the topological skeleton are computed by formerly rewriting the directorfieldasasetofordinarydifferentialequations(odes): d x dt = n ˆ (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 (x c,y c,z c ) are such that n ˆ ( x c ) = 0. We can further investigate the structure of the trajectories close to the stationary point by examining the Jacobianmatrixofthepartialderivativesofthedirectorfield, Jˆ n = n i x j with (x j x, y,z).if nature materials 3

4 thismatrixisnon singular,thestationarypoint x c (x c,y c,z c )isacriticalpoint. 51 Theeigenvalues andeigenvectorsofthejacobiandescribethelocalbehavioraroundacriticalpoint.wehave usedthenumericallibrarypfskel 53 tofindthepositionsofthecriticalpointsc 1 andc 2,located nearthetopandthebottomglassplates,respectively.bothc 1 andc 2 haveonerealeigenvalue and two complex conjugate eigenvalues. The two eigenvectors corresponding to the complex eigenvaluesdefineaplaneroughlyparalleltotheglassplateswhilethedirectionassociatedto therealeigenvalueisroughlyalongthe ˆ z axis.thecriticalpointsarehyperbolicandthefact thatwehavetwocomplexconjugateeigenvaluesindicatesthatthedirectorfieldspiralsaround these defects. 51 We have computed the streamlets (flow lines representing the motion of a massless particle) tangent to the director field. The streamlets originate very close from the criticalpoint(defect)locationsandmovealongtheeigendirectionsofc 1 andc 2.Equationsof motionaresolvedusingeithera2 nd ora4 th orderrunge 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 approachesforminimizationofelasticfreeenergy )comparedtotheexperimentalstudyof directorfieldconfigurationsaswellastopologicalanalysisfortheexperimentally observedt3 2sandT3 3swillbereportedelsewhere. SUPPLEMENTARYREFERENCES Numberingcontinuesfromthereferencelistinthemaintextofthemanuscript. 49.Allender,D.W.,Crawford,G.P.,andDoane,J.W.Determinationoftheliquid crystalsurface elasticconstantk 24.Phys. Rev. Lett.67, (1991). 4 nature MATERIALS

5 50.Polak,R.D.,Crawford,G.P.,Costival,B.C.,Doane,J.W.andZumer,S.Opticaldetermination ofthesaddle splayelasticconstantk 24 innematicliquidcrystals.phys. Rev. E49,R978(1994). 51. Asimov, D. Notes on the topology of vector fields and flows. Tech. Rep., NASA Ames Research Center.RNR (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 curveskeletonsof3dobjects.the Visual Computer21, (2005). 54.Anderson,J.E.,Watson,P.E.&Bos,P.J.LC3D:Liquidcrystaldisplay3 Ddirectorsimulator softwareandtechnologyguide(artechhouse,boston,2001) 55.Gil.,L.J.Numericalresolutionofthecholestericunwindingtransitionproblem. J. Phys. II France5, (1995) 56.Sonnet,A.Kilian,A.&Hess,S.Alignmenttensorversusdirector:Descriptionofdefectsin nematicliquidcrystals.phys. Rev. E52, (1995) nature materials 5

6 SUPPLEMENTARYFIGURES FigureS1.ComputersimulationsandFCPMimagingofthein planecross sectionsofthet3 1 structure.a,schematicrepresentationofthecellwiththehyperbolicpointdefectsshownby bluedotsandthetwistescapednon singulardisclinationringshownbyaredline.b,computer simulated nˆ ( r ) inthecentralplane(b bcross sectionshownina)ofthet3 1structurecoplanar withthedisclinationring.c,f,in planecross sectionsof nˆ ( r ) inthevicinityofpointdefectsnear thebottomplate,c,andthetopplate,f.d,g,correspondingsimulatedande,h,experimental FCPM textures. The red bars in e and h indicate the corresponding FCPM linear polarization states. 6 nature MATERIALS

7 FigureS2.FCPMimagingoftheT3 2swithdifferentlocationsanddiametersofdisclination rings. a c, FCPM vertical cross sections of three different T3 2s in a cell of thickness d = 15µm withdifferentdiametersofthedisclinationringatthebottomsurface.d,e,3dfcpm images of the T3 2 structures having the disclination rings d at the bottom and e at the top surface. nature materials 7

8 FigureS3.Computer simulatedverticalcross sectionsofthet3 1structure.a,Thestructure consistsoftheradialtwistof inthecentralplaneofthecellaswellastwopointdefects closetothesubstrates;theinsetsshowsimulated inmutually orthogonalxzandyzcrosssections intersecting the hyperbolic point defects at the top and bottom glass plates. b, Computer simulateddirectorfieldintheaxialcross section. 8 nature MATERIALS

9 Figure S4. Elastic free energy and the reduced diameter of the T3 1 configuration as a functionofd/p.a,freeenergyofahexagonalarrayoft3 1scomputedforaunitcellvolumeof 38µm 3 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 differentcholestericfingerscf1andcf2 18,19 (thedensityiscalculatedforthevolumeoftheunit cell of a 2D hexagonal lattice formed by Torons). b,normalized equilibrium diameter L / p of thet3 1structureasafunctionof d / p. nature materials 9

Interaction between colloidal particles in chiral nematic liquid crystals

Interaction between colloidal particles in chiral nematic liquid crystals Manoj Bhushan andey, I. Smalyukh Department of hysics, Liquid Crystal Materials Research Centre, University of Colorado at Boulder,