An equilibrium rotator glass-forming phase for long-ranged repulsive colloidal rods

Domain: biology
Published: 2026-03-18
Editorial outcome: minor_revision

Abstract
Glasses,amorphoussolidphasesnearlyalwaysoutofequilibrium,remain
poorlyunderstooddespiterecentprogress.Hereweshowbyquantitativereal-spaceexperimentsandcomputersimulationstheexistenceofanequilibrium
glass-formingphasewhichformsduetoasubtleinterplaybetweentherota-tionalandtranslationaldegreesoffreedominasystemofchargedcolloidal
rods.Inthisrotationalglass-formingphase,thepositionalcoordinatesare
glass-like,whiletherotationsremainliquid-like.Thisphasecanbereversibly
switchedintoacrystallinesolidthroughafirst-orderphasetransitionwith
minimalparticlerearrangementsbyanexternalelectricfield.Wespeculate
thatthisrotatorglass-likephaseformsduetotheanisotropicparticleinter-actionsathighervolumefractions,destabilizingthecrystal.Findinganequi-libriumglassyrotatorphasewillleadtonewinsightsonhowtranslationsand
rotationsaffectphasebehavior,includingglassformationand,additionally,
allownewtheoreticalapproachestobeusedtostudytheglasstransition.

Introduction
The important differences in the properties between liquids and
coordinateshavethedynamicsofliquidscanflowandhencearecalled
crystallinesolidscanbeexplainedbydifferencesinboththeirstruc- liquidcrystals3,5.Theabilityofsuchphasestorotatethepolarizationof
ture and dynamics. In terms of structure, particles in liquids have lightisusedinnematicliquidcrystalsdisplays6.
short-rangedpositionalandorientationalorder,whileforcrystalsboth Glasses are yet another important and intriguing class of
arelong-ranged1.Withrespecttothedynamics,incrystalsthelong- materials: they are solids whose structure resembles that of a
timeself-diffusioncoefficientsforbothtranslationsandrotationsof liquid,butwheretheparticles’long-timeself-diffusioncoefficients
particlesaretypicallyzeroonexperimentaltimescales,whileinliquids areassmallasforcrystals7–18.Theyareoftendescribedasliquids
bothdisplaysignificantmobility.Additionalstatesofmatterarepos- thathavesolidifiedbeforetheirstructurebecomescrystallineand
sible—oftenassociatedwithanisotropicshapedparticlesormolecules, thus are out of equilibrium7–13. However, not all non-crystalline
likerodsorplates—wheresomeofthedegreesoffreedomassociated solidsareinherentlyoutofequilibrium.Forexample,particlesthat
witheitherthepositionalororientationalorderanddynamicsarelike canformasmallnumberofflexiblebondswiththeirneighborscan
thoseofcrystals,whiletheremainingdegreesoffreedombehavelike form an arrested disordered equilibrium state at low
thoseofliquids1–3.Suchphaseshaveinterestingmixesofproperties. temperatures19,20, although for different reasons than the phase
Forinstance,inrotatorcrystalssomeoralloftherotationaldegreesof discussedinthepresentwork.
freedomareliquid-like,whilethepositionalorderisthatofasolid2,4. In this work, we demonstrate the existence of an equilibrium
Therotationsofthemoleculesweakenthestrengthofsuchsolidsso rotatorglass-likephase.Thisphaseexhibitstwointriguingproperties:
muchthat they candeform under theirown weight, leading to the a combination of liquid-like rotations and translational dynamics
name“plasticcrystals”2.Ontheotherhand,phaseswhere(someof)the whichareneardynamicalarrest,aswellasbeinganequilibriumphase
rotational degrees of freedom are solid-like, but the positional withshort-rangedpositionalorder.Weexplorethisphaseusingboth
real-spaceexperimentsonchargedcolloidalrod-shapedparticles4,5,21,22
andcomputersimulations.

Results
Westudiedasystemoffluorescentlylabeled,micron-sized,charged
silicarodswithashaperesemblingspherocylinders4,5,21,22characterized
byanend-to-endlengthl,diameterdandaspectratiol/d~4(Fig.1a).
Therodsweredispersedinanindexmatchingsolventwhichreduces
thevanderWaalsattractionstovaluessmallcomparedtothethermal
energyk T.Additionally,theparticleswerecoatedwithshortligand
B
moleculeswhichprovidethemwitharepulsiveinteractionthatissteep
andshort-rangedwithrespecttotheroddiameterandlength.Finally,
therangeofthecharge-basedrepulsionsischaracterizedbytheDebye
screeninglengthκ−1,whichisdirectlyrelatedtotheionicstrengthof
thesuspendingfluid.Whentheionicstrengthisrelativelylargeand
henceκ−1isshort(Fig.1b),theparticlesformthephasesknownforhard
rods,whichforl/d>3.5aresmecticLCphases3,5,22.Theseare1Dsolids,
with2Dliquidlayersofparallelorientedrods(Figs.1c,d).Conversely,
atlowionicstrength,therepulsionsaredominantovertheparticle
shapeandwhenκ−1becomescomparabletol(Fig.1e)rotatorcrystal
(or plastic crystal) phases can be formed4. Figure. 1f,g shows an
exampleofabody-centered-cubic(BCC)plasticcrystal(viewedfrom
BCC(110)plane).Forκl=0.8,i.e.,atascreeninglengthsimilartothe
particle length, the rods still form a rotator crystal phase at inter-
mediatedensities.However,whentherodswerecompressedtohigh
enoughdensities,thesystemfailedtocrystallize,andinsteadformeda
‘rotatorglassyphase’(Fig.1i,j).Thisphaselackslong-rangedpositional
order,highlightedbytheFFTintheinsetFig.1j.Additionally,inthis
phasetheparticlesarestronglycagedbytheirneighbors,leadingto
glass-like translational motion, whereas rotational motion remained
almostfree.ThisisshowninFig.1jbythesuperpositionof500images,
measuredoveratime-intervalΔt=163τ0,withτ0thetimefortherod
todiffuseoveritsowndiameteratinfinitedilution.
Togetamorequantitativepictureofthedynamicsofthisphase,
weanalyzedparticletrajectoriesfrom3Dconfocalmicroscopyimage-stacksasafunctionofincreasingvolumefractionϕ(seeSupplemen-taryFig.1andSupplementaryMov.1),andcalculatedthetranslational
and rotational self-correlation functions and analyzed the resulting
relaxation times (for details see the Methods section and Supple-mentaryFig.2)aswellasthemean-squareddisplacements(Supple-mentaryFig.3a,b).Figure2ademonstratesthatforavolumefraction
ofϕ≈0.017,thetranslationaldiffusionsloweddownbytwoordersof
magnitude,whiletherotationaldiffusionremainedmuchmorecon-stant.Atthisvolumefraction,particletrajectorieswerealmostcom-pletelycagedfortheduration oftheexperiment(seeFig.2bfora
typical particle trajectory and Supplementary Fig. 3c for a larger
ensemble)whilewealsoclearlyobserveddynamicalheterogeneities
(seeSupplementaryFig.4),ahallmarkofasystemapproachingthe
glass transition23. At the highest volume fraction we investigated
(ϕ>0.020),rotationsalsostartedtofreezein(Fig.2a–b).
To explore the origin of this rotator glass-like phase, we per-formedcomputersimulationsandfree-energycalculationsonasim-plifiedmodelforchargedcolloidalrods.Tothisend,weemployeda
Yukawasegmentmodel24,inwhichachargedrodisdividedintoinn
segmentsofequallydistributedpointchargesalongtherodaxis,see
Fig.2c.Theinteractionsbetweentworodsisthengivenbythesumof
theinteractionsbetweenthesegments,seeMethods.Notethatinthe
modelsystemtheinteractionsaresignificantlyshorter-ranged(κl=1.8)
thanintheexperimentstomakethesimulationsmorefeasible.Spe-cifically,thischoiceensuresalloursimulationsneededtoprobethe
phasediagramareeitherfluidorcrystal,andhencecanbeequilibrated
withoutissue.Theequilibriumphasebehaviorofthesesoftrod-like
particleswasdeterminedbycalculatingtheHelmholtzfreeenergyof
boththefluidphaseandtheBCCrotatorcrystalphase,asdescribedin
theMethods.ThephasediagramisshowninFig.2d.Inagreementwith
experiments,wefindthatforsufficientlychargedrods(highenough
Γ),thesystemundergoesafluidtoaBCCrotatorcrystalphasetran-sitionatintermediatedensitiesfollowedbyare-entrantBCCrotator
crystaltofluidtransitionathigherdensities.Hence,thesefree-energy
calculationsclearlysuggestathermodynamicoriginforare-entrant
disorderedphaseinthelesslong-rangedandapproximatedcomputer
simulationsmodel.There-entrantstabilityofthedisorderedphasecan
beexplainedbythefrustrationsinducedbytheeffectivecorrelated
orientationstheparticlesundergoasthesystemiscompressed.Atlow
andintermediatedensities,theparticlesarefarapartduetothecharge
repulsion,andasaresulttheinteractionsaremoreisotropic.Incon-trast, at higher densities, the closer proximity of particles to their
neighborsimpliesthattheshapeoftheparticleplaysanimportantrole
intheinteractions.Asaresult,differentpairsofparticlesfeeleffec-tively different interactions based on their orientations, causing an
effective polydispersity in the interactions.This would lead to frus-tration,awellknowndriverofglassformation12,25.
Toinvestigatethedynamicsofthisre-entrantdisorderedphase,
weperformedBrowniandynamicssimulationsoftheYukawasegment
model26. We calculated the meansquare displacement〈Δr2(t)〉 for a
wide range of densities ρ and extracted the long-time translational
diffusioncoefficientD,whichwerendereddimensionlessbynormal-t
izationwithD0 (SupplementaryFig.5).Atsufficientlyhighdensities,
the diffusivity decreased exponentially with increasing density. This
highlights,asexpected,agradualarrestofthedynamicsasthedensity
isincreased.Althoughinthemodelsystemtheinteractionsaresig-nificantlyshorter-ranged,thereisgoodqualitativeagreementbetween
simulation and experiment (compare also Fig. 2e with Supplemen-taryFig.1).
To further illustrate experimentally the thermodynamic origin
andnatureofthephaseneardynamicalarrest,weappliedanexternal
high-frequencyACelectricfieldthatalignedtheparticleswiththefield
direction.WeshowinFig.3(seealsoSupplementaryFig.6andSup-plementary Mov. 2,3)that subsequently the rotatorglass-like phase
transformedintoa3Dcrystal.Thefieldrestrictedtherotationsofthe
rods,andasaresultitloweredtheanisotropyoftheparticleinterac-tionsallowingthemtoformaBCClatticestretchedinthefielddirec-tion(seealsoSupplementaryFig.7).Oncetheelectricfieldwasturned
off,particlesregainedtheirorientationalfreedomandthecrystalline
phaserevertedbacktoarotatorglassyphase,whichdemonstratesthat
thisglass-formingamorphousphasehasalowerfreeenergythanthe
crystallinephase.Theradialdistributionfunctiong(r)(Fig.3f)shows
thelackoflong-rangedorderinthenear-arrestedrotatorphase.
Westudiedthereversibilityoftherotatorglassyphasetocrystal
transitionbychangingtheelectricfieldstrengthinsmallerandslower
steps.AsshowninFig.4,weobservedstronghysteresiseffects,indi-cative ofafirst-orderphasetransition.Particlesquicklyaligned and
thenstartedtoorderpositionallymoreslowly(Fig.4a–c).Thesample
initiallybecamepolycrystallineoverlargerdistances(~100μm),but
afterapproximately1hlargeordereddomainswerepresent(Fig.4d).
Whenthefieldstrengthwasthendecreasedagain,stronghysteresis
was observed in both the rotational dynamics and positional order
(Fig.4h–i).WhenthefieldstrengthdroppedbelowapproximatelyE=
45V/mm,the(rotator)crystallinelatticebecameunstable,andcom-pletelyreturnedtotheoriginalrotatorglass-likestate.Whenthe
electricfieldwasincreasedanddecreasedatmuchfasterrates(Sup-plementaryFig.6),weobservedthesamephasesatthestartandendof
the procedure, supporting that these phases are independent of
quenchspeed.
For an electric field strength E = 90 V/mm and average inter-particle spacing r = 1.4 l we calculated that the strengthofthedipole-dipoleinteractionismuchsmallerthank T4.Thisdemonstratesthat
B
the contributionof dipole-dipoleinteractions to the stability of the
crystalphasecanbeneglected.Theapproximately33%elongationof
theBCClatticealongthefielddirectionmustthereforebecompletely
due to the anisotropy of the particle and its electrostatic repulsive
potential.
Wefurtheranalyzedthe3DstructuresbycalculatingtheVoronoi
cellsaroundeachparticle(SupplementaryFig.8)andfoundthatthe
rotatorphaseneardynamicalarresthadonaverage14nearestneigh-bors and Voronoi faces with on average 5 edges, representing the
numberofparticlessurroundinganear-neighborbond.Interestingly,
thesetwodistributionsarealmostidenticaltothedistributionsofa
glassconsistingofhardspheres17orsphereswithalong-rangerepul-sivepotential27.Wefurtherconcludethatthisphasetransitionoccurs
with minimal particle rearrangements, since the number of nearest
neighborsremained constant during the crystallization and melting
process. Finally, we determined the distribution of the Voronoi cell
volumes (Supplementary Fig. 8e). Using the hard-core particle
dimensions (as measured with TEM), both averaged values for the
glassandthecrystalcorrespondtoavolumefractionofϕ=0.017.This
value is higher than the range of volume fractions where a plastic-crystalphasewasfoundinearlierwork4.

Methods
Particlesynthesis
Fluorescentsilicarod-likeparticlesweresynthesizedasdescribedin
ref.22.Theparticlesweredyedusingfluoresceinisothiocyanate(FITC,
isomer I, 90%, Sigma-Aldrich). The particles consisted of a three
layeredstructure:anon-fluorescentcore,anFITClabeledshell(ca.40
nm)andanon-fluorescentoutershell(ca.100nm).Inthesecondstep,
wemodifiedthesilicarodswithoctadecyltrimethoxysilane(OTMOS,
90%, Sigma-Aldrich) by an ultrasonic-assisted coating procedure. In
brief,asolutionofOTMOS,butylamine(BA,99.5%,Sigma-Aldrich)and
toluene(1:1:10v/v/v)wasprepared,andthen10wt%ofsilicarods,that
haddbeendriedundernitrogenflow,wereaddedtothissolution.The
suspensionwassonicatedat30–55°Cfor4h(Branson2250).After-wards,thecolloidalrodswerewashedinturnwithtoluene,cyclohex-aneandcyclohexylchloride(CHC, >98%,Merck).Finally,thecolloidal
rodsweredispersedindeionizedCHCforfurtheruse.Theparticles
hadanend-to-endlengthl=2.29μm(δ=6.0%),diameterd=600nm(δ
=6.5%),andaspectratiol/d=3.8.Here,landdarethemeanlengthand
diameterandδisthepolydispersity(standarddeviationdividedbythe
mean). A second silica rod system was used without OTMOS mod-ification, which had an average end-to-end length l = 2.6 μm (poly-dispersityδ=9%),diameterd=630nm(polydispersityδ=6%)and
aspectratiol/d=4.1.

SamplepreparationandElectricfieldsetup
The particles were suspended in cyclohexyl chloride (CHC). The as
receivedCHC(conductivity ≫1.000pS/cm)wasdeionizedbyusing
molecular sieves (0.4nm, Aldrich) and activated aluminum oxide
(Aldrich).Afterpurification,theCHChadaconductivityaslowas5–10
pS/cm.Inthissolvent,theparticlesacquiredanegativecharge(surface
potential−70mV)andaDebyescreeninglengthκ−1 ~5μm.
TheDebyescreeninglengthκ−1,whichisgivenby:
κ−1= ϵ r ϵ 0 k B T / 2N Ie2
A
withϵ thedielectricconstantofthesolvent,ϵ thepermittivityof
r 0
vacuum,N theAvogadroconstant,Itheconcentrationofionsinthe
A
suspensionandetheelementarycharge.
The sample was concentrated to a volume fraction ϕ ~0.02 by
eithersedimentationundergravityorbycentrifugationwithamax-imumof60g.Rectangularfusedquartzcapillaries(0.1×2mm2and
0.2 ×2 mm2, VitroCom, UK) were used as sample cells. Electrodes
consistingofa3nmlayerofCranda6–10nmlayerofAuweresputter-coatedontwoopposingoutersurfacesofthesecapillaries.Thermo-couplealloywires(diameter50μm,Goodfellow)wereconnectedto
theelectrodeswithsilverpaint(SPI-paint).Theendsofthewireswere
wrappedaroundstandardelectronicwiresthatinturnwereconnected
totheelectricalset-up.Weusedafunctiongenerator(Agilent33120A)
to generate a sinusoidal signal with a frequency of 1 MHz and an
amplitudeof2.0V(peak-to-peak).Thissignalwassenttothesample
viaawidebandamplifier(Krohn-Hite,7602M)usedtovarythefield
strengthinthesample.Thefieldstrengthwasmeasuredwithadigital
phosphor oscilloscope (TektronixTDS3052). We applied a high fre-quencyACfieldtopreventpolarizationoftheelectricdoublelayerof
theparticles.Afterfillingthecapillaries,theywerefixedonaglassslide
andsealedwithUV-curedglue(Norland,No.68).

Electricfieldcalculation
Becausetheelectrodeswereontheoutsideofthecapillaries,there
werethreelayersofmaterialbetweenthetwoelectrodes.Inthiscase,
thefieldstrengthisgivenby5
E= V ϵ1ϵ2ϵ3 / (ϵ0 d1 ϵ2 ϵ3 + d2 ϵ1 ϵ3 + d3 ϵ1 ϵ2)
withVtheappliedvoltage,ϵ thedielectricconstantofthesample,
0
ϵ 1,−3 thedielectricconstantoflayers1-3andd 1−3 thethicknessoflayers
1-3. In our experiments, layer 1 and 3 are the glass capillary walls,
(thicknessd ~0.1mmandϵ=3.5)andlayer2is(primarily)thesolvent
CHC(thicknessd ~0.1mmandϵ=7.6).Thefieldstrengthisthusgiven
byE=1.9×10−3V(V/μm).Allfieldstrengthsinthischapteraregivenin
unitsV /μm.
RMS
Inthesteppedelectricfieldexperiment,weincreasedtheelectric
fieldstrengthinstepsof22.5V/mm.Thedurationofeachstepwas
2.8minandthetimeintervalbetweenmeasurementswas4min.

Confocalmicroscopymeasurements
Particles were imaged using a confocal microscope (Leica SP8)
equipped with a fast 12 kHz resonant scanner and a GaAsP hybrid
detector(LeicaHyDTM).Imageswith8-bitpixel-depthwereacquired
using a white light laser with a selected wavelength of 488 nm. A
confocalglycerolimmersionobjective63x/1.3(Leica)wasused,which
is optimized for refractive index n = 1.47. To avoid hydrodynamic
interactionswiththewall,particleswereimagedatleast20μmdeep
intothesample.Toinvestigatethe3Dstructureofthesuspensions,
datastackswererecordedwithtypicaldimensionsof512×256×120
pixels3 with voxel size 160 ×160 ×321nm3, corresponding to
82×41×39μm3,whichwererecordedwithaframe-rateof0.3fps.To
trackthe3Dparticlemotion,data-stackswereacquiredwithtypical
dimensionsof512×65×25pixels3withvoxelsize160×160×321nm3,
corresponding to a volume of 82 ×10 ×8 μm3. The frame-rate for
recordingthese3Dvolumeswastypically4fps.2Dconfocalmicro-scopy images had dimensions of 512 ×512 pixels2 and pixel size of
150×150nm2,correspondingtotypically75×75μm2.The2Dimages
wererecordedwithaframe-rateof2–20fps.

Analysisofstaticstructure
Radialdistributionfunction.Todeterminetheaveragestaticstruc-tureinoursamples,wecalculatedtheradialdistributionfunctiong(r).
Foragivenparticle,thisfunctiondescribestheprobabilityoffinding
anotherparticleadistancerawayandisgivenby
g(r) = 1/ρ2 < Σi Σj≠i δ(r)δ(r - rij) >
withδtheDiracdeltafunction,ρthebulknumberdensityandthe
angularbracketsdenotinganensembleaverage.Nisthetotalnumber
of particles in the system. The distribution is normalized such
thatg(r→∞)=1.
Nematicorderparameter.Toquantifythe3Dorientationalorderwe
calculatedthe3Dnematicorderparameterdefinedby
S = 3 <cos2θ> - 1 / 2
3D
withθtheanglebetweenthemainaxisoftherodubandthenematic
directornb.WefindS andnbbycalculatingthelargesteigenvalueand
3D
correspondingeigenvectorofthestandard3×3nematicorderpara-metertensor
Q3D = 1/N Σi=1 to N (3uiαuiβ - δαβ) / 2
withu iαtheα-componentoftheunitvectorpointingalongthemain
axisofparticleiandα=x,y,z.
Hexagonal bond order parameter. To quantify the 2D positional
orderweusedthelocalhexagonalbond-orientationalorderparameter
ψ givenby
6k
ψ 6k = 1/n(k) Σj=1 to n(k) exp(i6θ(rjk))
withn(k)thenumberofneighborsofparticlek(whichistakentobe
thenumberofparticlesthatarewithinacertaincut-offdistancer of
c
theparticle),r thevectorconnectingparticlekandits neighborj,
jk
θ(r )theanglebetweenr andanarbitraryreferenceaxisandiinthe
jk jk
exponenttheimaginaryunit.Inaperfecthexagonallayer,theangles
θ(r )aremultiplesof60°andjψ j=1.
jk 6k
Voronoicellconstruction.TheVoronoicellofaparticlecontainsall
thepointsinspacethatareclosertotheparticlethantoanyother
particleinthesystem.Formonodisperseparticlesthesecellsconsistof
faces and straight edges, called polyhedra. We used a publicly
availableVoronoisoftwarelibrary(Voro++30)tocalculate3DVoronoi
cells,thenumberofVoronoinearestneigbours,thenumberofedges
perVoronoifaceandtheaverageVoronoicellvolume.

Analysisofparticledynamics
Tostudytheparticledynamics,weappliedourparticle-fittingalgo-rithm to time-series of 3D confocal microscopy data-stacks, as
described in detail in ref. 21. We calculated the mean squared dis-placement(MSD)givenby
Δr2(t) = <|r(t) - r(0)|2>
whichwefittedtotheexpression
Δr2(t) = 6D t + 6ϵ2
t t
with D the rotationally averaged translational diffusion coefficient
t
and ϵ the error in measurement of each of the coordinates of the
t
particle31. For the highest volume fractions, we couldnotreach the
regionwherethelong-termtranslationaldynamicsbecomediffusive
again,i.e., 〈Δr2(t)〉 ~ t. For the mean squared angular displacement
(MSAD)weusedtheexpression32,33
<Δub2(t)> = 2[1 - (1 - ϵ2)exp(-2D t)]
r r
withD theaveragerotationaldiffusioncoefficientandϵ the mea-r r
surementerrorinthedeterminationofub(t).Tofurtherquantifythe
rotationalmotionoftheparticles,wealsocalculatedtheorientation
auto-correlationfunction
C(t) = <ub(0)·ub(t)>
whichwefittedwithastretchedexponentialfunction,givenby
C(t) = (1 - ϵ2)exp(-(t/τ )β)
r r
withϵ themeasurementerrorinthedeterminationofthedirectionof
r
themainaxisoftherods,τ thetypicalorientationrelaxationtimeand
r
βtheKohlrauschexponent.Atinfinitedilutionwehaveβ=1andτ =1/
(2D ).Fortherotationalandtranslationaldynamicswealsocalculated
r
theself-intermediate scatteringfunction,givenby
Fs(q,t) = < 1/N Σj=1 to N exp[iq·(rj(t) - rj(0))] >
withqthewavevectorwithmagnitudeequaltothefirstpeakofthe
radialdistributionfunctiong(r),andr(t)thethreedimensionalposi-tionoftherod.Theself-intermediate scatteringfunctionwasfitted
with
Fs(t) = a exp(-(t/τ )β)
t
withaapre-factorthatincludestheerrorinthedeterminationofthe
positionoftherod,τ thetypicaltranslationalrelaxationtimeandβthe
t
Kohlrauschexponent.

Yukawasegmentmodel
WeemployedtheYukawasegmentmodel24,inwhichachargedrodis
dividedintoinnsegmentsofequallydistributedpointchargesalong
therodaxis.Theinteractionsbetweentworodsisthengivenbythe
sumoftheinteractionsbetweenthesegments:
βUij(ri - rj, ui, uj) = βΓ Σa,b=1 to n exp(-κrab) / rab
with κ the screening length, βΓ the energy scale between twoseg-ments,andβ=1/k T,wherek istheBoltzmannconstantandTisthe
B B
temperature,and
rab = |ri - rj + ud (2a-1-n)/2 - ud (2b-1-n)/2|
ij i n j n
thedistancebetweensegmentaofrodiandsegmentbofrodj,where
d denotesthedistancebetweenneighboringsegments
n
d = l / sqrt((n+1)(n-1))
n
Sincethesystemweinvestigatedishighlycharged,therodsnever
came into close contact and we can safely omit the hard-core
interactionsinoursimulations.

Freeenergycalculations
Wedeterminedtheequilibriumphasebehaviorofourrod-likeparti-clesbycalculatingtheHelmholtzfreeenergyofboththefluidphase
andtheBCCcrystalphase.Tothisend,wefirstcalculateda“reference"
free energy of boththefluidandcrystalphaseatasingledensityusing
theWidominsertionmethodandFrenkel-Laddmethod,respectively34.
Subsequently, the Helmholtz free energy of both phases wascalcu-latedforarangeofdensitiesandenergyscales(temperatures)using
thermodynamicintegrationoftheequationofstateandtemperature-dependentaveragepotentialenergy,ascalculatedusingMonteCarlo
simulations in the NVT-ensemble. In these calculations, we have
restrictedourselvestodensitiesandenergyscalesuptoρl3≈0.65and
βΓ≈68,respectively.Notethatinthesimulationswehavechosento
useashorterrangedinteractionthanintheexperiments.Thischoice
hasensuredthatthesystemisfastenoughforthesimulationsinvolved
withthethermodynamicintegrationtoequilibrate.This,however,also
meansthatthesystemshowsare-entrantfluid,notare-entrantglass.

Browniandynamicssimulations
WeperformedBrowniandynamicssimulationsoftheYukawasegment
model following ref. 26. Here, we have mapped the self-diffusion
coefficients to the experimental system by matching the ratios
betweentherotationalandthetwotranslationaldiffusioncoefficients.
We measured time in units of the translational diffusion at infinite
dilutionτ =l2/D0.
t

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Report for “An Equilibrium Rotator Glass for Long-Ranged Repulsive Colloidal Rods” by Besseling et al.\nThis manuscript presents what seem at first glance to be interesting results for an equilibrium rotator glass. The concept\ninvolves a re-entrant transition to an amorphous state at high concentration, which the authors interpret as an equilibrium\nglass. I think that the experiments are quite interesting. However the manuscript in its present form is far from appropriate for\nNature Communications.\nThe dynamical regime accessible to such large colloids is very limited. The authors report they have measured dynamics\nover 163 times the time for the rod to diffuse its own diameter. Two orders of magnitude change in dynamics though is far\nfrom sufficient to term a material a glass.\nIn the seminal work of Brambilla et al (Phys. Rev. Lett. 102, 085703 (2009)), 7 orders of magnitude of change in relaxation\ntime were recorded, and still the sample was not termed a glass. (I am surprised that the authors do not cite this paper). Now\nit is true that the particles here are longer, but still the authors lack experimental evidence for a glass by some considerable\nmargin. Of course it is 14 orders of magnitude for a molecular glass.\nThat in itself, might be OK, as they perform computer simulations to back up their claims. Yet, strangely no details of the\nrelaxation protocols used in the simulations are given. This seems odd, since the simulations surely need to be equilibrated\nat each step of the thermodynamic integration and some measure must be employed to ensure that this is correctly done. To\nbe clear, mapping the experimental 2 orders of magnitude increase in relaxation time would only correspond to a\ntemperature of perhaps 0.55 in the well-known Kob-Anderson Binary Lennard-Jones model (for example), yet in that\ncommunity, T=0.55 is only considered a weakly supercooled liquid. Certainly not a glass. It is almost certainly possible\ntechnically to perform “thermodynamic integration” with those kinds of dynamics, as there is a separation between the short-\ntime and the full “alpha” relaxation. But that doesn’t make such a material a solid glass. And one would certainly question the\nmeaning of such thermodynamic integration.\nWith no evidence as to the equilibration of the simulations, it is entirely unclear what the high-density amorphous state is.\n(Longer experiments would also be more than helpful, but I appreciate that these can be harder to perform than the\nsimulations).\nSo the authors can either revise their manuscript in line with the evidence they have: that is, evidence for an intriguing retract\namorphous state whose dynamics are too slow for their experiments to clarify, ie it might be a glass but (more likely, given\nthe lack of “equilibrium glasses” in nature), a more viscous liquid.\nOr if they can perform simulations where they show categorically, that they have obtained an equilibrium free energy for this\nstate then there could be something to discuss. For example SWAP dynamics can be used to probe systems well below the\nmolecular glass transition of 14 orders of magnitude change in relaxation time (eg Ninarello et al Phys. Rev. X 7, 021039\n(2017)). So if the states found here can be shown convincing not to decay under SWAP dynamics, then indeed the authors\ncould claim to have identified glass.\nFinally a small point. Please remove the comment about volume fraction 0.59 or at least explain exactly what is meant. Is it\nthe mode-coupling value of 0.58 that is referred to? This was shown by the paper mentioned above to be a dynamical\ncrossover. Or are the authors thinking of some limit of the (possible) “true” glass transition, one prediction from the same\npaper being 0.614, but most lying at a volume fraction above 0.61.", "reviewer_2": "The physical mechanisms that lead to arrested states of matter, especially gels and glasses, are of paramount importance in\nseveral branches of science, such as Physics, Biology, Chemical Engineering, Materials Science, among others.\nIn this manuscript, authors have reported on the existence of a new “equilibrium” glassy state here defined as rotator glass\nmade up of colloidal spherocylinders that interact with long-ranged repulsive forces. By means of real space measurements\nand both Monte Carlo and Brownian dynamics simulations, authors provide partial evidence on the role that the\nanisotropic particle interactions might have on the destabilization of the crystal phase at high volume fractions. Authors discuss how the\ninterplay between translational and rotational degrees of freedom has on the colloidal phase behavior.\nI consider that the results obtained by the authors are original and of crucial importance for the understanding of the\nconnection that exists between different types of degrees of freedom on both the structure and dynamics of many-body\nsystems close to dynamical arrest. This topic is appropriated for Nature Communications due to its scientific implications for\nthe vast community of Condensed Matter. Furthermore, I found the paper well written, discussed and organized. However, to\nrecommend the acceptance of the manuscript, authors should properly address the following points:\n1. Most of the arguments to understand the appearance of the rotator glassy state is based on the interplay between the\ntranslational and rotational degrees of freedom, and that interplay is discussed in terms of the long-time self-diffusion\ncoefficient and the typical rotational and translational trajectories of individual particles when the concentration is increased.\nHowever, this information provides partial evidence of the existence of such a glassy state. Nevertheless, it is possible to\nmeasure (and calculate) other dynamical observables that will elucidate the particle relaxation and the dynamical coupling\nbetween degrees of freedom, see, e.g., Soft Matter 16, 170 (2020). I then suggest the authors to determine, report and\ndiscuss the translational and rotational self-correlation functions and from those quantities it is possible to extract the\nrelaxation times, which will elucidate whether this state corresponds or not to a particular class of glassy state.\n2. I still do not understand why the colloidal dispersion relaxes back to the rotator glass state from the crystalline-like\nconfiguration; the latter is obtained when the electric field is switched on. This is completely physically counterintuitive since\nthe crystal configuration is, in principle, the state of minimum energy. I then wonder whether such a “equilibrium” glassy state\nis not a dynamical arrested state. This point should be discussed in more detail to clearly state that authors found a new non-\nequilibrium thermodynamic state.", "reviewer_3": "Solutions of charged rod-shaped colloids are studied by confocal\nmicroscopy. The effective aspect ratio can be changed by varying the screening\nlength which allows the observation of different equilibrium and glassy structures.\nWhile liquid crystalline and plastic crystal phases are known, a novel rotational\nglass is observed, where translations are frozen and disordered while rotations remain liquid-like.\nWith increasing volume fraction of the colloidal rods, the ratio of translational to\nrotational diffusion coefficients decreases by more than a factor ten.\nApplying an electric field, the rods can be aligned which enables positional crystalline order\nto form. Switching off the field, the particles again become disordered and\nreturn to the rotational glass state. Brownian dynamics simulations of a simplified\nmodel are performed to rationalize the experiments, and observe fluid and plastic\ncrystal states.\nThis is a very interesting study of a peculiar glass state which, in my opinion, is\nof broad interest to the Materials Science and Soft Matter community.\nThe study nicely combines experiments and simulations, and the paper is very well written\nand informative. Still, there are a couple of points that should be addressed before I can\nrecommend publication.\nThere are theoretical predictions by Schilling and Scheidsteger [1]\non orientational glasses and glass transitions of slightly elongated particles.\nThese should be discussed carefully. My impression is that the experiments see the\nglass states predicted by Schilling and coworkers. Moreover, the predictions could\nbe tested more qualitatively and quantitatively by looking at orientational correlation\nfunctions for different angular momenta. It would also be interesting to compare\nthe experimental effective aspect ratio to the value of the critical aspect ratio predicted in [1].\nThere are detailed simulations by Chong, Kob and colleagues [2,3] testing the\norientational glass predictions. These simulations and observations should be\ndiscussed carefully including with respect to the present simulations. The present\nsimulations do not observe glassy states in contrast to the experiments, while the\nolder simulation saw the orientational glass formation.\nThe simulations appear to observe two equilibrium phases, fluid and plastic crystal.\nThe experiments on the other hand find glassy states. This difference should be stated\nclearly and should be discussed. Why do the simulated states reach equilibrium, while\nthe experimental samples arrest?\nConnected with this discussion the presentation of the experimental states as\nequilibrium solid state (abstract line 22) versus 'rotational glass' (line 26) should\nbe re-assessed. The experiments observe glass states that form upon increasing the\ndensity or, at high enough density, upon decreasing an aligning electric field.\nYet, this does not convincingly prove that the rotational glass states are equilibrium\nstates in my opinion. It would require to check. e.g. that these states do not age and\nare independent of the quench speed or other parameters of the preparation history.\nFig 3 f: One of the g(r) lines is shifted, or? Stating this would be helpful.\nIn lines around 208 of the main text, it is stated that the rotator glass re-formed\nupon decreasing the electric field. In the caption of Fig. 4 f the opposite is stated\n('crystalline order still persisted'), and Fig 4f itself is inconclusive. The description\nneeds to be corrected, and the Steinhard order measure (Fig 4i) should be discussed\nin order to quantify the difference between glass and crystal.\n[1] R. Schilling and T. Scheidsteger, “Mode coupling approach to the ideal glass\ntransition of molecular liquids: Linear molecules,” Phys. Rev. E 56, 2932 (1997).\n[2] S.-H. Chong and W. Kob, “Coupling and decoupling between translational\nand rotational dynamics in a supercooled molecular liquid,” Phys. Rev. Lett. 102,\n025702 (2009).\n[3] S.-H. Chong, A. J. Moreno, F. Sciortino, and W. Kob, “Evidence for the weak\nsteric hindrance scenario in the supercooled-state reorientational dynamics,”\nPhys. Rev. Lett. 94, 215701 (2005)."}, "author_response": "Response to reviewers:\nReviewer #1:\nThis manuscript presents what seem at first glance to be interesting results for an\nequilibrium rotator glass. The concept involves a re-entrant transition to an amorphous\nstate at high concentration, which the authors interpret as an equilibrium glass. I think that\nthe experiments are quite interesting. However the manuscript in its present form is far\nfrom appropriate for Nature Communications.\nThe dynamical regime accessible to such large colloids is very limited. The authors report\nthat they have measured dynamics over 163 times the time for the rod to diffuse its own\ndiameter. Two orders of magnitude change in dynamics though is far from sufficient to term\na material a glass.\nIn the seminal work of Brambilla et al (Phys. Rev. Lett. 102, 085703 (2009)), 7 orders of\nmagnitude of change in relaxation time were recorded, and still the sample was not termed\na glass. (I am surprised that the authors do not cite this paper). Now it is true that the\nparticles here are longer, but still the authors lack experimental evidence for a glass by\nsome considerable margin. Of course it is 14 orders of magnitude for a molecular glass.\nThat in itself, might be OK, as they perform computer simulations to back up their claims.\nYet, strangely no details of the relaxation protocols used in the simulations are given. This\nseems odd, since the simulations surely need to be equilibrated at each step of the\nthermodynamic integration and some measure must be employed to ensure that this is\ncorrectly done. To be clear, mapping the experimental 2 orders of magnitude increase in\nrelaxation time would only correspond to a temperature of perhaps 0.55 in the well-known\nKob-Anderson Binary Lennard-Jones model (for example), yet in that community, T=0.55 is\nonly considered a weakly supercooled liquid. Certainly not a glass. It is almost certainly\npossible technically to perform “thermodynamic integration” with those kinds of dynamics,\nas there is a separation between the short-time and the full “alpha” relaxation. But that\ndoesn’t make such a material a solid glass. And one would certainly question the meaning of\nsuch thermodynamic integration.\nWith no evidence as to the equilibration of the simulations, it is entirely unclear what the\nhigh-density amorphous state is. (Longer experiments would also be more than helpful, but I\nappreciate that these can be harder to perform than the simulations).\nSo the authors can either revise their manuscript in line with the evidence they have: that is,\nevidence for an intriguing retract amorphous state whose dynamics are too slow for their\nexperiments to clarify, ie it might be a glass but (more likely, given the lack of “equilibrium\nglasses” in nature), a more viscous liquid.\nOr if they can perform simulations where they show categorically, that they have obtained\nan equilibrium free energy for this state then there could be something to discuss. For\nexample SWAP dynamics can be used to probe systems well below the molecular glass\ntransition of 14 orders of magnitude change in relaxation time (eg Ninarello et al Phys. Rev.\nX 7, 021039 (2017)). So if the states found here can be shown convincing not to decay\nunder SWAP dynamics, then indeed the authors could claim to have identified glass.\nFinally a small point. Please remove the comment about volume fraction 0.59 or at least\nexplain exactly what is meant. Is it the mode-coupling value of 0.58 that is referred to? This\nwas shown by the paper mentioned above to be a dynamical crossover. Or are the authors\nthinking of some limit of the (possible) “true” glass transition, one prediction from the same\npaper being 0.614, but most lying at a volume fraction above 0.61.\nResponse to reviewer #1:\nWe thank the referee for the thoughtful comments on our manuscript and have revised our\nmanuscript based on additional findings as well as based on the concerns raised by the referee.\nBelow is a point-by-point explanation and response to the referee’s comments:\nWe would like to emphasize that our system is extremely soft, evidenced by the extremely\nlarge Debye screening length. It is known that there can be up to 7 orders of magnitude\ndifference in the relaxation times for a soft matter system compared to a molecular system,\nhowever, even for a soft system our system is extremely soft, resulting in relaxation times\nincreasing much more gradually than typical colloidal systems. Regarding the concern of the\norders of magnitude of the relaxation times, (see also the comments by Referee #2), we have,\nbased on additional experimental results, calculated the translational and rotational self-\ncorrelation functions and analyzed the resulting relaxation times as a function of density up\nto the highest packing fraction we could obtain (where both translational motion and\nrotational motion is completely frozen in), see Figure 2a and SI Figure 2 in the revised\nmanuscript. We have added the details of the calculation of the correlation functions to the\nmethod section on page 6, line 518-529.\nIn the region where most of the experiments were performed (incl the electric-field switching\nexperiments), there is a very strong decoupling of the translational and rotational degrees of\nfreedom, however, there are still limited particle rearrangements taking place (reminiscent of\ncage escape observed in glassy states of nearly hard-sphere colloids, i.e. particles with much\nsteeper and more short-ranged interactions), resulting in a system that is not completely\nfrozen-in on the timescales of our experiments. However, we also found that these phases\nare very close to a volume fraction where all motion (translation and rotations) completely\nfreezes in (see Figure 2a and SI Fig. 3), and observe (again in agreement with hard-sphere\nsystems) dynamical heterogeneities, a hallmark of glassy dynamics (see SI Fig. 4). We therefore\nchanged the term ‘equilibrium rotator glass’ and refer to this phase in the paper as an\n‘equilibrium rotator glass-forming phase’, or ‘glass-like phase’, indicating an amorphous phase\nthat is close to dynamic arrest. We refrain from using the terminology ‘supercooled liquid’ as\nthis implies an underlying stable crystal phase which, intriguingly, in our system of\nmonodisperse particles was not found and likely does not exist, given also the simulation\nresults. Furthermore, we do not refer to this phase anymore as a ‘solid’.\nWe would like to emphasize that the conclusion that this glass-forming phase near dynamic\narrest is an equilibrium phase still holds and this is in our view an important finding. In our\nexperiments we have found that the rotator glass-forming phase can be crystallized by\napplication of an AC electric field that aligns the particles in the field direction. When the field\nis turned off again, the particles do not remain on their crystal lattice but return to the\ntranslationally disordered (and nearly arrested) phase. This serves as experimental evidence\nthat the induced crystalline state is not the equilibrium phase but that it is the glass-like\namorphous phase that has lower free energy. This finding is supported with computer\nsimulations of soft rods interacting with simplified interactions but with the same competing\ntendencies. Furthermore, we observed experimentally strong hysteresis in the system when\nswitching between the crystalline phase and the amorphous phase (Fig. 4), which is indicative\nof a first-order equilibrium phase transition. We have further clarified this point, by adding an\nadditional sentence in the main paper, see line 220-221.\nWith respect to how we ensured that the simulations reached equilibrium, in order to study\nthe phase behaviour, we intentionally focused on a system with a slightly shorter interaction\nrange. One consequence of this choice was that the re-entrant phase was disordered, but not\nstrongly glassy. As a result, we were easily able to equilibrate the system and thus obtain the\nfree energies of the system. Therefore, simulation tricks to speed up equilibration were not\nneeded. While we carefully only referred to the re-entrant phase as disordered, it was not\nclearly stated that this was the case. Hence, we now clarify this both in the main text (line\n164) and in the methods section (line 560).\nFurthermore, we thank the referee for notifying us about the work of Brambilla et al., which\nwe have added as a reference to the introduction section (line 76, ref 18).\nWe agree with the referee that the comment about the 0.59 volume fraction is not strongly\nexplained nor supported by further analysis, so we have removed this comment as suggested\nby the referee (we removed line 272 –276).\nAll (other) additions or modifications we have made are highlighted in blue in the manuscript,\nand text that has been removed is crossed out.\nReviewer #2 :\nThe physical mechanisms that lead to arrested states of matter, especially gels and glasses,\nare of paramount importance in several branches of science, such as Physics, Biology,\nChemical Engineering, Materials Science, among others.\nIn this manuscript, authors have reported on the existence of a new “equilibrium” glassy\nstate here defined as rotator glass made up of colloidal spherocylinders that interact with\nlong-ranged repulsive forces. By means of real space measurements and both Monte Carlo\nand Brownian dynamics simulations, authors provide partial evidence on the role that the\nanisotropic particle interactions might have on the destabilization of the crystal phase at high\nvolume fractions. Authors discuss how the interplay between translational and rotational\ndegrees of freedom has on the colloidal phase behavior.\nI consider that the results obtained by the authors are original and of crucial importance for\nthe understanding of the connection that exists between different types of degrees of\nfreedom on both the structure and dynamics of many-body systems close to dynamical\narrest. This topic is appropriated for Nature Communications due to its scientific\nimplications for the vast community of Condensed Matter. Furthermore, I found the paper\nwell written, discussed and organized. However, to recommend the acceptance of the\nmanuscript, authors should properly address the following points:\n1. Most of the arguments to understand the appearance of the rotator glassy state is based\non the interplay between the translational and rotational degrees of freedom, and that\ninterplay is discussed in terms of the long-time self-diffusion coefficient and the typical\nrotational and translational trajectories of individual particles when the concentration is\nincreased. However, this information provides partial evidence of the existence of such a\nglassy state. Nevertheless, it is possible to measure (and calculate) other dynamical\nobservables that will elucidate the particle relaxation and the dynamical coupling between\ndegrees of freedom, see, e.g., Soft Matter 16, 170 (2020). I then suggest the authors to\ndetermine, report and discuss the translational and rotational self-correlation functions and\nfrom those quantities it is possible to extract the relaxation times, which will elucidate\nwhether this state corresponds or not to a particular class of glassy state.\n1. We agree with the referee to calculate the translational and rotational self-correlation\nfunctions. Based on additional experimental data, for a range of volume fractions, we analyzed\nthe resulting relaxation times as a function of density up to the highest packing fraction we\ncould obtain (where both translation motion and rotational motion is completely frozen in),\nsee Figure 2a and SI Figure 2 in the revised manuscript. We have added the details of the\ncalculation of the correlation functions to the method section on page 6, line 518-529.\nIt is clear from this analysis, that in the region where most of the experiments were performed\n(incl the electric-field switching experiments), there is a very strong decoupling of the\ntranslational and rotational degrees of freedom, however, there are still limited particle\nrearrangements taking place (reminiscent of cage escape observed in glassy states of nearly\nhard-sphere colloids, i.e. particles with much steeper and more short-ranged interactions),\nresulting in a system that is not completely frozen-in on the timescales of our experiments.\nHowever, we also found that these phases are very close to a volume fraction where all\nmotion (translation and rotations) completely freeze in (see Figure 2a and SI Fig. 3), and\nobserve (again in agreement with hard-sphere systems) dynamical heterogeneities, a hallmark\nof glassy dynamics (see SI Fig. 4). We therefore changed the term ‘equilibrium rotator glass’\nand refer to this phase in the paper as an ‘equilibrium rotator glass-forming phase’, or ‘glass-\nlike phase’, indicating an amorphous phase that is close to dynamic arrest. We refrain from\nusing the terminology ‘supercooled liquid’ as this implies an underlying stable crystal phase\nwhich, intriguingly, in our system of monodisperse particles was not found and likely does not\nexist, given also the simulation results. Furthermore, we do not refer to this phase anymore\nas a ‘solid’.\n2. I still do not understand why the colloidal dispersion relaxes back to the rotator glass\nstate from the crystalline-like configuration; the latter is obtained when the electric field is\nswitched on. This is completely physically counterintuitive since the crystal configuration is,\nin principle, the state of minimum energy. I then wonder whether such a “equilibrium” glassy\nstate is not a dynamical arrested state. This point should be discussed in more detail to\nclearly state that authors found a new non-equilibrium thermodynamic state.\n2. In our experiments we have found that the rotator glass-forming phase can be crystallized\nby application of an AC electric field that aligns the particles in the field direction. When the\nfield is turned off again, the particles do not remain on their crystalline lattice but return back\nto the translationally disordered (and nearly arrested) phase. We agree with the referee that\nthis is indeed counterintuitive. However, it serves as experimental evidence that the induced\ncrystalline state is not the equilibrium phase but that it is the glassy amorphous phase that has\nlower free energy. This finding is supported with computer simulations of soft rods interacting\nwith simplified interactions but with the same competing tendencies. Furthermore, we\nobserved strong hysteresis in the system when switching between the crystalline phase and\nthe amorphous phase (Fig. 4), which is indicative of a first-order equilibrium phase transition.\nTo clarify this point, we have added an additional sentence in the main paper, see line 220-\n221.\nAll other additions or modifications we have made are highlighted in blue in the manuscript,\nand text that has been removed is crossed out.\nReviewer #3:\nSolutions of charged rod-shaped colloids are studied by confocal\nmicroscopy. The effective aspect ratio can be changed by varying the screening\nlength which allows the observation of different equilibrium and glassy structures.\nWhile liquid crystalline and plastic crystal phases are known, a novel rotational\nglass is observed, where translations are frozen and disordered while rotations remain\nliquid-like.\nWith increasing volume fraction of the colloidal rods, the ratio of translational to\nrotational diffusion coefficients decreases by more than a factor ten.\nApplying an electric field, the rods can be aligned which enables positional crystalline order\nto form. Switching off the field, the particles again become disordered and\nreturn to the rotational glass state. Brownian dynamics simulations of a simplified\nmodel are performed to rationalize the experiments, and observe fluid and plastic\ncrystal states.\nThis is a very interesting study of a peculiar glass state which, in my opinion, is\nof broad interest to the Materials Science and Soft Matter community.\nThe study nicely combines experiments and simulations, and the paper is very well written\nand informative. Still, there are a couple of points that should be addressed before I can\nrecommend publication.\nThere are theoretical predictions by Schilling and Scheidsteger [1]\non orientational glasses and glass transitions of slightly elongated particles.\nThese should be discussed carefully. My impression is that the experiments see the\nglass states predicted by Schilling and coworkers. Moreover, the predictions could\nbe tested more qualitatively and quantitatively by looking at orientational correlation\nfunctions for different angular momenta. It would also be interesting to compare\nthe experimental effective aspect ratio to the value of the critical aspect ratio predicted in\n[1].\nWe agree with the referee that this is indeed a very important paper to cite in our manuscript,\nand have now included it in the discussion (line 309-316, ref 28). Note that we are unable to\nmake a direct comparison to a “critical aspect ratio” from that paper as the critical behaviour\nwas examined for dipolar hard spheres, which is quite different from the system we study in\nthis paper. Nonetheless, as we now write in the discussion, we qualitatively see the same\ntransitions.\nThere are detailed simulations by Chong, Kob and colleagues [2,3] testing the\norientational glass predictions. These simulations and observations should be\ndiscussed carefully including with respect to the present simulations. The present\nsimulations do not observe glassy states in contrast to the experiments, while the\nolder simulation saw the orientational glass formation.\nThe simulations appear to observe two equilibrium phases, fluid and plastic crystal.\nThe experiments on the other hand find glassy states. This difference should be stated\nclearly and should be discussed. Why do the simulated states reach equilibrium, while\nthe experimental samples arrest?\nTo probe whether the disordered phase could be fully an equilibrium phase, we intentionally\nused a model that was close to the experiments but that was less slow (further from the\nglass). We found that simply by using a model with a slightly shorter interaction range, the\nsystem sped up sufficiently to be able to make fully equilibrium measurements (e.g. of the\nequation of state and the integration path associated with Einstein integration). As a positive\nconsequence of this choice, we were able to draw an equilibrium phase diagram, however, a\nmore negative consequence was that the simulation model had significantly faster dynamics\nthan the experimental system.\nConnected with this discussion the presentation of the experimental states as\nequilibrium solid state (abstract line 22) versus 'rotational glass' (line 26) should\nbe re-assessed.\nBased on additional experimental results we calculated the translational and rotational self-\ncorrelation functions and analyzed the resulting relaxation times as a function of density up\nto the highest packing fraction we could obtain (where both translation motion and rotational\nmotion were completely frozen in), see Figure 2a and SI Figure 2 in the revised manuscript.\nWe added the details of the calculation of the correlation functions to the method section on\npage 6, line 518-529.\nIn the region where most of the experiments were performed (incl the electric-field\nswitching), there is a very strong decoupling of the translational and rotational degrees of\nfreedom, however, there are still limited particle rearrangements taking place (reminiscent of\ncage escape observed in glassy states of nearly hard-sphere colloids, i.e. particles with much\nsteeper and more short-ranged interactions), resulting in a system that is not completely\nfrozen-in on the timescales of our experiments. However, we also found that these phases\nare very close to a volume fraction where all motion (translation and rotations) completely\nfreeze in (see Figure 2a and SI Fig. 3), and observe (again in agreement with hard-sphere\nsystems) dynamical heterogeneities, a hallmark of glassy dynamics (see SI Fig. 4). We therefore\nchanged the term ‘equilibrium rotator glass’ and refer to this phase in the paper as an\n‘equilibrium rotator glass-forming phase’, or ‘glass-like phase’, indicating an amorphous phase\nthat is close to dynamic arrest. We refrain from using the terminology ‘supercooled liquid’ as\nthis implies an underlying stable crystal phase which, intriguingly, in our system of\nmonodisperse particles was not found and likely does not exist, given also the simulation\nresults. Furthermore, we do not refer to this phase anymore as a ‘solid’.\nThe experiments observe glass states that form upon increasing the\ndensity or, at high enough density, upon decreasing an aligning electric field.\nYet, this does not convincingly prove that the rotational glass states are equilibrium\nstates in my opinion. It would require to check. e.g. that these states do not age and\nare independent of the quench speed or other parameters of the preparation history.\nIn our experiments we found that the rotator glass-forming phase can be crystallized by\napplication of an AC electric field that aligns the particles in the field direction. When the field\nis turned off again, the particles do not remain on their crystalline lattice but return to the\ntranslationally disordered (and nearly arrested) phase. This serves as experimental evidence\nthat the induced crystalline state is not the equilibrium phase but that it is the glass-like\namorphous phase that has lower free energy. This finding is supported with computer\nsimulations of soft rods interacting with simplified interactions but with the same competing\ntendencies. Furthermore, we observed experimentally strong hysteresis in the system when\nswitching between the crystalline phase and the amorphous phase (Fig. 4), which is indicative\nof a first-order equilibrium phase transition. We have further clarified this point, by adding an\nadditional sentence in the main paper, see line 220-221.\nRegarding the quench speed, we performed similar experiments as shown in Fig. 4 but with\nmuch longer than the 4 min time intervals in between the measurement (results not shown\nhere) as well as much faster quench rates (see SI Fig. 6). In all cases, we observed the same\nphases at the start and end of the quench procedure, supporting that these phases are\nindependent of quench speed. We have added further clarification about this in the main\npaper, see line 239 – 243.\nFig 3 f: One of the g(r) lines is shifted, or? Stating this would be helpful.\nIn Fig. 3f, the g(r) line of the rotator glass-forming phase is indeed shifted upwards for\nclarity, this is added in the figure caption.\nIn lines around 208 of the main text, it is stated that the rotator glass re-formed\nupon decreasing the electric field. In the caption of Fig. 4 f the opposite is stated\n('crystalline order still persisted'), and Fig 4f itself is inconclusive. The description\nneeds to be corrected, and the Steinhard order measure (Fig 4i) should be discussed\nin order to quantify the difference between glass and crystal.\nBased on the comment of the referee we changed the text in the caption of Fig. 4 for\nclarification. We have added in the caption that at zero field strength the system relaxed back\nto the amorphous glass-like phase (Fig. 4f). The remark ‘crystalline order still persisted’ was\nto highlight that the sample was still crystalline for field strength 45 V/m in the down ramp so\nwe changed the text to highlight this comment applies only to panel d and e.\nThe fact that the phase in Fig. 4f is no longer in crystalline state can be inferred from the value\nof the average local hexagonal bond-order parameter which is < 0.4.\nWhile the value of the average local hexagonal bond-order parameter in a perfect hexagonal\nlattice is 1.0, it is difficult to assign a clear threshold value for a crystal without assessment of\nthe long-ranged order (i.e. bond-order orientation correlation function). Please note that in\nour case, because we needed to image the system in 3D with high enough frame-rate to follow\nthe rotations of the particles, the view-of-view in Fig. 4 is limited, especially in the z-dimension,\nwhich is why we have used a 2D bond-order parameter as an approximation (clearly showing\nthe difference between the phase with and without field, as well as the hysteresis). For a\nsnapshot of the full structure and 3D analysis of the crystalline structure see Fig. 3 and SI Fig.\n7, which was recorded at the same field strength as Fig. 4d (90 V/mm).\nAll (other) additions or modifications we have made are highlighted in blue in the manuscript,\nand text that has been removed is crossed out.\n[1] R. Schilling and T. Scheidsteger, “Mode coupling approach to the ideal glass\ntransition of molecular liquids: Linear molecules,” Phys. Rev. E 56, 2932 (1997).\n[2] S.-H. Chong and W. Kob, “Coupling and decoupling between translational\nand rotational dynamics in a supercooled molecular liquid,” Phys. Rev. Lett. 102,\n025702 (2009).\n[3] S.-H. Chong, A. J. Moreno, F. Sciortino, and W. Kob, “Evidence for the weak\nsteric hindrance scenario in the supercooled-state reorientational dynamics,”\nPhys. Rev. Lett. 94, 215701 (2005)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Second report for “An Equilibrium Rotator Glass for Long-Ranged Repulsive Colloidal Rods” by Besseling et al.\nThe authors have done a good job in revising their manuscript in line with the comments made during the first round of\nreviewing, and I am happy to recommend publication of the revised manuscript. As noted in the first round of reviewing, this\nis an interesting result that would not be expect a prior and as such, it is suitable for the broad readership of Nature\nCommunications.", "reviewer_2": "I really appreciate the effort made by the authors to include all the points raised in my previous report. The results obtained\nby the authors are original and of crucial importance for\nthe understanding of the connection that exists between different types of degrees of freedom on both the structure and\ndynamics of many-body systems close to dynamical arrest. I therefore recommend the publication of the manuscript in its\npresent form.", "reviewer_3": "In response to the three referee reports, the authors have added further data and analysis, and have extended their\ndiscussion. I find the paper now warranting publication in Nature communications as it reports solid insights into an\nintriguing novel glassy state. The concept of equilibrium amorphous state will certainly motivate theory and simulations.\nOverall the study shows that anisotropy can lead to novel insights into the formation of amorphous solids with intriguing (de-)\ncoupling of rotational and translational motion which is an actively developing reasearch area."}, "author_response": ""}]

Decision letter