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SUPPLEMENTARY INFORMATIONdoi: 10.1038/nmat2592
nature materials | www.nature.com/naturematerials 1
1
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:
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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SUPPLEMENTARY INFORMATION doi: 10.1038/nmat2592
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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
nature materials | www.nature.com/naturematerials 3
SUPPLEMENTARY INFORMATIONdoi: 10.1038/nmat2592
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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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SUPPLEMENTARY INFORMATION doi: 10.1038/nmat2592
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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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SUPPLEMENTARY INFORMATIONdoi: 10.1038/nmat2592
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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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SUPPLEMENTARY INFORMATION doi: 10.1038/nmat2592
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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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SUPPLEMENTARY INFORMATIONdoi: 10.1038/nmat2592
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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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SUPPLEMENTARY INFORMATION doi: 10.1038/nmat2592
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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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SUPPLEMENTARY INFORMATIONdoi: 10.1038/nmat2592
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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 / .