Static andDynamicProperties ofTwoDimensional CoulombClusters Biswarup Ash,1 J. Chakrabarti,2 and Amit Ghosal1 1Indian Institute of Science Education and Research-Kolkata, Mohanpur, India-741246 2S.N.BoseNational CentreforBasicSciences, Block-JD,Sector-III,Salt Lake, Kolkata-700098 Westudythetemperaturedependence ofstaticanddynamic responsesof Coulombinteractingparticlesin two-dimensionaltrapsacrossthethermalcrossoverfromanamorphoussolid-toliquid-likebehaviors. While static correlations, that investigate the translational and bond orientational order in the confinements, show thefootprints of hexatic-like phase at low temperature, dynamics of theparticles slow down considerably in 7 thisstate–reminiscentofasupercooledliquid. Usingdensitycorrelations, weprobeintriguingsignaturesof 1 long-livedinhomogeneitiesduetotheinterplayoftheirregularityintheconfinementandlong-rangeCoulomb 0 interactions. Therelaxationatmultipletimescalesshowstretched-exponentialdecayofspatialcorrelationsin 2 irregulartraps. Temperaturedependenceofcharacteristictimescales,depictingthestructuralrelaxationofthe system,showstrikingsimilaritieswiththoseobservedfortheglassysystemsindicatingthat,someofthekey n signaturesofsupercooledliquidsemergeinconfinementswithlowerspatialsymmetries. a J 9 I. INTRODUCTION temindeedhelpsinprobingtheintricaciesofdifferentthermal phases. Thedynamicalresponseof2Dsystems, close to the ] n liquid-hexatictransition,showsomestrikingsimilaritieswith Long-rangeinteractingclassicalparticlesindisorderedme- n thatofliquidsclosetotheglasstransition[25].Moleculardy- dia can give rise to exotic phases, e.g, hexatic glass [1], - namicssimulationof2Dsystemofcolloidparticles[26]and s which is characterized by short range positional order and i experiment with granular materials [27, 28] show that with d quasi-longrange orientationalorder. This is quite similar to theincreaseinorientationalorder,thedynamicsoftheparti- . the usual hexaticphase in two dimension(2D), as discussed t clesslowdownandbecomeheterogeneous.Particlesundergo a in Kosterlitz, Thouless [2, 3], Halperin, Nelson, and Young m ‘cagingeffect’(i.e.,gettingtemporarilytrappedinthecageof (KTHNY)[4–6]meltingscenario.Thoughatruehexaticglass its neighbors) and cooperative dynamics. These results into - inthethermodynamiclimitattractedcriticaldebates[7,8],it d sub-diffusive behavior of the mean square displacements as may be realized in finite systems [9]. However, there have n observedinexperiments[29]andsimulations[30–32]. beenattemptsmadetounderstandexperimentswithcolloidal o While true phase transitions pertain to bulk systems only, c particles[10],binary-alloy[11],magneticbubblelattice[12] forfinitesystemsthenotionofsolid-andliquid-like‘phases’ [ anddisorderedtypeIIsuperconductors[1]inlightofahexatic havebeenusedsuccessfullytocharacterizequalitativebehav- glass. 1 iorsofmacroscopicsystems [33]. In recentyears, static and v Confined systems, with small number of particles, are of dynamicpropertiesof2Dfiniteclusterswerestudiedfordif- 8 great current interests due to their controlled experimental ferent types of confinements [34–38] and interaction poten- 3 tunability over a wide range of parameters. Such systems tials[35, 39, 40] acrossthe thermalmelting. These theoreti- 3 are not only significant from technologicalperspective, they 2 calstudiesaremotivatedbyexperimentalrealizations,suchas are also very important for fundamental Physics: They are 0 colloidalsuspensions[41–43],confinedplasma[44–46],elec- idealplaygroundforexploringcomplexinterplayofinterac- . tronsinquantumdotsinhighmagneticfields[47,48],radio- 1 tionsanddisorder.Inthispaper,wedevelopanunderstanding 0 frequencyiontraps[49]andelectronsonthesurfaceofliquid ofsuchinterplaybystudyingstaticanddynamiccorrelations 7 helium [50]. Recent studies of static [35] and dynamic [37] innano-clusterswith(long-range)Coulomb-interactingparti- 1 properties of Coulomb interacting particles in irregular ge- : cles. ometryhave confirmedthe thermalcrossoverfrom‘solid’ to v i Standardorder-disordertransitionsdonotoccurin2D[13] ‘liquid’-like phases. From the study of spatio-temporalcor- X because,atrueorderedphaseisweakenedbyfluctuationspro- relations [37] in such system, it is found that with decrease r ducingonlyquasi-orderedstate.Stillatransitionfromsolidto inT someofthekeysignaturesofglassydynamicsemerges. a liquidoccurswithaninterveninghexaticphasewhichischar- For example, as system approachessolidity from liquid-like acterizedbyshortrangepositionalorderandquasi-longrange phase,thedynamicsoftheparticlesbecomeslowandhetero- bond-orientationalorder.Suchtwostepmeltingscenario,pre- geneous.Isthisasignatureofahexatic-glassphase? dictedbytheKTHNYtheoryforclean2Dsystems,isrealized Weaddressthisbroadquestionbypresentinginthispaper incertainexperiments[14–16]andsimulations[17,18]. The theresultsofextensivecomputersimulationsonthestaticand possibilityofextendingsuchideasinthepresenceofdisorder dynamicpropertiesofCoulombinteractingparticlesindiffer- is still a subjectof currentresearch [19–22]. Doesthe melt- entconfinementgeometries.Fromouranalysisofstaticprop- ingofCoulombinteractingparticlesin2Dsystemcorroborate erties,wefindthatwhilethepositionalorderisdepletedeven with the KTHNY melting, even in the absence of any disor- at the lowest T for an irregular trap, a solid-like phase can der? Inspiteofthirtyyearsofresearch,thereisnodefinitive stillbeidentifiedatlowT duetothebondorientationalorder. answer,tothebestofourknowledge[23]. A liquid-like phase emerges[35] with the breakingdown of Study of dynamics across the melting often provides cru- bondorientationalorderbeyondacrossovertemperatureT , X cialadditionalinsights[24].Thedynamicalbehaviorofasys- however, the inhomogeneities in the liquid persists up to a 2 muchlargertemperature–aboutanorderofmagnitudelarger correspondingly the time scale will also be renormalized to thanT . Beyondthislargetemperaturescale( 10T ),the t′ = ~φ−2/3α−1/3t. Suchrenormalizationoflength,energy X X ∼ systemcrossesovertoastandardisotropicliquid. Ourresults andtimewilleventuallymaketemperaturetobeexpressedas are indicative of a hexatic-like low T phase in the confine- T′ =E′/k ,wherek istheBoltzmannconstant.Inorderto B B ments. Finally, the analysis of the trajectories of individual haveanestimate ofthesenewlengthandtimescalesin con- particlesandtheT-dependenceofrelaxationtime-scalesshow ventionalunits,weconsiderelectronsinGaAsheterostructure characteristicsquitesimilartothoseobservedforglassform- withatypicalconfinementenergyof~ω =1meV.Theabove 0 ersneartheglasstransitiontemperature[25]. scaled length (r′), energy (E′) and time (t′) unit takes the Therestofthepaperisorganizedasfollows: InSec. IIwe valueof630A˚,1.7meVand376fs,respectively.Inextracting will give the details of the models and methods used in our thesevalueswehaveused,q = e,thechargeofanelectron, − simulationsemphasizingthewaydisorderisintroducedinfi- m=0.067m ,themassofanelectronandǫ=13[54]. e nitesystems. Wewillalsodiscussaboutthemethodsusedto VCr isquadraticinlength-scalewhileVIr isquartic. In conf conf study the system. Then we present, in Sec. III, our results ordertofacilitateajustifiedcomparisonbetweenthetwo,we for static propertieswhere we analyze the thermal crossover needtoexpressthequarticirregularconfinementintheunits throughthetemperaturedependenceofbondorientationalor- ofquadraticcircularconfinement. Thisisachievedbysetting der. In Sec. IV dynamicpropertiesare analyzed to identify, the parameter a = mω2/2r′2 a′, which includes both ω 0 0 in particular,any signatureof a glass-like behavior, and also anda scalingfactora′, forirregularconfinement. Thusa ju- to study the temperature dependenceof structural relaxation dicious choice of a′,(cid:0)which now(cid:1)controls the strength of the time(s). Finally,weconcludeinSec. V. irregularconfinement, brings the two confinementson equal footing on dimensional ground. Following the same scale transformationsasforthecircularcase,weobtainthefollow- II. MODELANDMETHOD ingHamiltonianforVIr : conf We consider N classical particles each with charge q in a x4 confiningpotentialV (r). Theseparticlesinteractvialong Ir = a′ +by4 2λx2y2+γ(x y)xyr conf H b − − range Coulomb potential and are restricted to move in 2D. (cid:20) (cid:21) TheHamiltoniandescribingsuchasystemreadsas: N 1 + (5) ~r ~r q2 N 1 N 1≤Xi 0.02, we get a very turesabimodalstructures,itspeaknearφ6 1(atlowT)signaling kl ∼ broad distribution signifying the breakdown of orientational solidityandtheoneatφ6 ≈0portrayingliquidity. order. Thus, the T-dependenceof P(ψ ) signals a thermal 6 | | crossoverfrom an orientationallyorderedsolid-like to a dis- withN =100-500describegenericbehaviorfeaturingquali- orderedliquid-likephaseinconfinedsystems. Itisinteresting tativelysimilarresults. to note that P(ψ ) shows hardly any thermal evolution for 6 | | The nature of melting by disrupting bond orientational T > 0.03. Thisisfoundtrueirrespectiveoftheconfinement order is usually tracked by studying the large r behav- geometry or total number of particles,N. This insensitivity ior of bond-orientational correlation function, g (r) = confirmsthatthescramblingoforienationalorderandhence 6 ψ (r)ψ∗(0) . However, for finite systems with small num- meltingiscompletebyT 0.03. h 6 6 i ≈ berofparticles,g (r)isnotveryuseful,becauseanytrendin WithdecreasingN (Fig.2(c)forN =150andFig.2(e)for 6 itsr-dependenceishardtodiscern[35]duetolimitedvalues N =75),wenoticeastrengtheningofasmallandfeeblepeak ofr. inP(ψ )near ψ = 1. We assertthatsuchpeakis spuri- 6 6 | | | | ous. Weverifiedthatthisiscontributedpredominantlybythe particlesontheboundary,havingveryfewnearestneighbors. Notethataparticlewithasoleneighbormustyield ψ =1, C. PROJECTIONOFBONDORIENTATIONALORDER 6 | | irrespective of the value of bond angle. Decrease in N in- creasestheratioofthenumberofboundary-tobulk-particles, In order to get additional insight into the degree of local enhancingthestrengthofsuchspuriouspeak. Fromourover- orientationalorder,we,next,lookintotheshortdistancepro- all study, we find that systems with N 100 tend to show jection of bond-orientationalcorrelation function. The mag- ≤ non-universalbehavior,notonlyforP(ψ )butalsoforother nitudeoftheprojectionofbondorientationalorder,φ (k),of 6 6 | | observables we describe below. On the other hand, systems particle k onto the average local bond orientational order of 5 0.3 lowT(=0.002),thedistributionissharplypeakedatφ 1 6 0.12 (a) VICronf (b) VCCronf N=75 (see Fig. 2(b)), depicting a strong correlation of the orie≈nta- χψ 0.2 NN==155000 tiinocnraelasoersd,esruincghocfoarrpealarttiioclnerwedituhcietssnaneadreastthniegihghTb(o=urs0..0A5s0T), 0.08 χψ the distribution becomes sharply peaked around φ 1. 6 ∼ N=75 Thus, for T > 0.02, the local six-fold order is lost even at 0.04 NN==155000 0.1 the second shell, representing a ‘liquid’-like state. We see that with decrease in N (Fig. 2(d,f)), the sharpness of the 0 .03 0 peak around φ6 1 gets considerably reduced at the low- ∼ est T(= 0.002) as the effect of the boundary particles be- (c) VICronf 0.6 (d) VCCronf comesmoreprominentforsmallersystemsize. ForN = 75 0.2 (Fig. 2(f)), an orientationalordering is hard to discern up to 0.4 thesecondneighbourevenatlowT. χφ χφ An earlier investigation [35] of IWM with N = 150 0.1 found T 0.02 (for our parameters) by studying the T- 0.2 X ≈ dependence of specific heat and Lindemann ratio, among otherquantities.HereweidentifyT bystudyingthefluctua- X 0 0 tionoftheorientationalorderparameters ψ andφ ,defined (e) VICronf (f) VCCronf as | 6| 6 0.8 0.8 0.6 0.6 N=75, fs χψ =N ψ6 2 ψ6 2 (8) f 0.4 f 0.4 NNN===715550, 00fl χφ =N(cid:2)hhφ| 26i|−ih−φ6hi|2 |i (cid:3) (9) (cid:2) (cid:3) Here,χ’sarethegeneralizedsusceptibilitycorrespondtoor- 0.2 0.2 der parameter ψ and φ , respectively. In Fig.3, we show 6 6 | | 0 0 0.01 0.02 0.03 0.04 0.05 0 0 0.01 0.02 0.03 0.04 0.05 thetemperaturedependenceofχψ (panela,b)andχφ (panel T T c,d)fordifferentsystemsizes. While,forabulksystem,sus- ceptibilitydivergesatthecriticalpoint,suchdivergenceturns FIG.3: T-dependenceofthegeneralizedsusceptibilitiesidentifying into a crossover in a finite system. The fluctuations in χ thecrossovertemperatureTX,forN = 500,150,75. Panel(a)and ψ and χ attain their maxima at the crossovertemperatureT (c)showsthethermalevolutionofχ andχ respectivelyinirreg- φ X ψ φ ular confinement. Whilethepeaks ofχ andχ sharpen withN, distinguishing the low temperature solid- and high tempera- ψ φ theirlocation,TX,sufferhardlyanychange.Panel(b)and(d)shows tureliquid-likephases,asseenfromFig.3(a-d). Thepeakat resultssimilartothoseinpanel(a)and(c)butforcirculartrap. The T 0.02 becomes sharper with increasing N (though for X ∼ fragileorderinthiscaseforN =75,degradesTX.Panel(e)and(f) N = 75, the circular confinementshows deviation), and we illustratethethermalevolutionofthesolid-andliquid-fractioninthe generallyfindthe valueof T quiteinsensitiveto the nature X systemforcircularandirregulartrap. Thecrossingofthetwotakes ofconfinement. Thus,T estimatedfrombondorientational placeinanarrowwindowofT (fordifferentN)forVcIornf(r),where order parameter is in goXod agreement with its previous esti- as,thetemperaturewindowisbroadforVcConrf(r). mates[35]. In order to compare T with the corresponding melting X itsnearestneighboursisdefinedas[63,64]: temperaturefor bulk2D systems, we estimate the parameter Γ(= √πn/T)atT = T . Itisfoundthatforbulksystems, X meltingoccursforΓ 137[55]. In Table I, we showΓ for 1 Nb differentN andfound≈thatΓremainscloseto137withinac- φ (k)= ψ (k)∗ ψ (l) (7) 6 (cid:12) 6 Nb 6 (cid:12) ceptabletoleranceforthelargeN’sofourstudy. (cid:12) Xl=1 (cid:12) (cid:12) (cid:12) While ψ (k) measur(cid:12)eshowtheneighborso(cid:12)fparticlekad- | 6 | (cid:12) (cid:12) N Irregular Circular just locally on a hexagonal environment, φ (k) determines 6 how the orientation of a particle fits into the mean orienta- n TX Γ n TX Γ tional field of its nearest neighbors. Thus, φ6(= φ6(k) for 75 2.42 0.017 0.003 162 1.90 0.011 0.001 222 k = 1,2, N)representsthebondorientationalcorrelation ± ± ··· 150 2.53 0.019 0.001 148 2.12 0.019 0.002 136 uptonextnearestneighbordistances. ± ± At low T, as most of the particles are surrounded by six 500 3.01 0.021 0.001 146 2.78 0.020 0.001 148 ± ± regularneighbors,andweexpectφ 1. WithincreasingT, 6 ∼ duetodepletedorientationalorder,φ6 decreasesandeventu- TABLEI:EstimationoftheparameterΓ=√πn/TX,characterizes allybecomesclosetozeroathighT. Thus,φ distinguishes the meltingin bulk 2D systems, for irregular and circular confine- 6 ahighT phaseclearlyfromalowT phase. ments. For bulk systems Γ 137 at the transition. Notethat the ≈ InFig.2(b,dandf)weshowthedistributionofφ atdiffer- averageparticledensity nisestimatedby computing theareacov- 6 entT forIWM with N = 500,150and75, respectively. At eredbyN particlesintherespectiveconfinements. 6 Since φ6 is a projection of ψ6, φ6(k) ψ6(k). Be- (a) N=150 (b) ≤ | | causemaximumvalueofeitherofψ orφ isunity,φ (k)+ 6 6 6 ψ (k) 2. Hence, we can estimate melting bydescribing r0 |eac6hpa|rt≤icletobeeither‘solid’-or‘liquid’-likeusingthema- = 0.85 1 2 2 jorityrule[63]:particlekissolid-likeifφ6(k)+|ψ6(k)|>1 y 3 y and liquid-like otherwise. In Fig. 3(e-f) we have shown the fractionofsolid-andliquid-likeparticlesatdifferentT forirregular and circular confinements, respectively. We see that atverylowT(= 0.002)almost90%oftheparticlesaresolid VICronf T = 0.006 likewhileathighT(=0.050)particlesaremostlyliquidlike. x x Interestingly,thetwocurvescrosseachotheraroundthesame (c) T asobtainedfromotherquantitiesabove. X Few points deserve special mention here: A phase with depletedpositionalorderbutfeaturingan orientationalorder in the scale of system size is indicative of a hexatic phase 3 within the KTHNY description of bulk 2D melting. How- y r(t)i (d) ever,insystemswithdisorder,liketheirregularconfinement considered here, similar characteristics should identify a so 1 r0 = 0.85 called ‘hexaticglass’ phase. This phase differsfromregular 2 3 hexatic phase in the sense that it supports both the thermal andquencheddefects,whereasthehexaticphaseinKTHNY x 0 3000 t 6000 9000 theoryallowsonlythermaldefects. While theexistenceofa (e) VICronf (f) VICronf thermodynamichexaticglassphase hasbeen debated[7], its signature in small finite systems, like IWM, can possibly be discernedasourresultsinprecedingsectionsuggest. T = 0.006 T = 0.006 t = 1000 t = 9500 y y IV. DYNAMICPROPERTIES ∆r/r0 ≤ 0.15 ∆r/r0 ≤ 0.15 0.15<∆r/r0 ≤ 0.50 0.15<∆r/r0 ≤ 0.50 Aslowandheterogeneousmotionofconstituentparticlesis 0.50<∆∆rr//rr00>≤11..00 0.50<∆∆rr//rr00>≤11..00 ahallmarkofglassysystems. Below,weresorttodynamical x x characterizationofdifferentphasesofoursystemin orderto explore its glassy nature, if at all. Some aspects of intrigu- FIG.4: (a) Initialpositions of sN = 150 particles(blue dots) in aspecificrealizationofanirregularconfinementalongwiththetra- ingslowandheterogeneousdynamicsinIWM,primarilyex- jectories of three particles at T = 0.006. Thick black lines indi- tractedfromtheanalysisofVanHovecorrelationfunction,are cate the net displacement of the particles over the duration of the recentlyreported[37]byus.Hereweexpandonthoseresults, simulation. (b-c) Illustrates the trajectories of the particles 2 and firstly,bycharacterizingqualitativelythemotionalsignatures 3 with larger displacements as shown in (a). Both of them jiggles inthetraps,andsubsequently,byextractingtimescalesasso- around their equilibrium position for a while and they make occa- ciated with the structural relaxation causing the depletion of sional and sudden jumps by distance r0. Particle 1, however, ∼ bondorientationalorder. doesnotmakesuchhops(notshownseparately). (d)t-dependence of the instantaneous distance ri(t) from the centre of the confinement for the three selected particles (i = 1,2,3). Displacement A. Analysisofthetrajectory (∆~ri(t) = ~ri(t) ~ri(0) )ofN = 500particlesinirregularcon- { − } finementisshownfor(e)t=1000and(f)t=9500atT =0.006. Focusingona specific realizationof the irregularconfinement, λ,γ = 0.635,0.20 ,we show in Fig. 4(a) the ini- { } { } tialconfigurationofN = 150particles(thickdots)afterMD and (c) (along with a solid line representing the displace- equilibration. On this we also superimpose the trajectory of ment). These particles jiggle around their equilibrium posi- threeparticles(indexedas1,2and3)duringthewholeMD- tion for certain time and then jump close to location of its simulation. Weseethat,atlowT,thenatureofthedynamics neighbor and spend some time before jumping to previous of individualparticles can be broadly classified in following (reversible motion) or a new location (irreversible motion). twocategories: Suchoccasionaljumpscanbecapturedbylookingintothein- (i) There are particles, like the one indexed 1 in Fig. 4(a), stantaneousposition, r (t) in Fig. 4(d) where i representthe i whichonly rattlesaroundtheir equilibriumpositionoverthe particle index. We see that within a sufficiently small time durationofwholeMDsimulation. window these particles change their positions moving by a (ii) Additionally, there are particles, indexed 2 and 3 in distance of the order of r . This motion can be regarded 0 Fig.4(a),whichmovelongerdistances(byseveralr )inthe as the cage breaking process and avalanches of such cage- 0 course of time. Their trajectories are enlarged in Fig. 4(b) breakingeventsgiverisetostring-likepatternforthedisplace- 7 (a) VIr N=500 (Gs(r,t)) [67] shows stretched exponential spatial decay 2.0 Conf (G (r,t) exp( lrk)). Here, we extend such studies to s kIr track the N∼-depen−dence of k(t) for different T as shown in 1.0 Fig. 5 (panel a: N = 500, panel b: N = 150 and panel c: 2.0 N = 75) for VIr . Here, we denote k for VIr and VCr 0.6 (a1) conf conf conf 1.0 T=0.020 askIr andkCr, respectively. Whilethe stretchedexponential 0.3 kIr TT==00..000026 TT==00..003500 tbiechualavri,otrhiesrgeecnoevreircyfoofrtahlel Nfin,atlhiesofitrnoepridceltiaqiulsiddliifkfeerb.eIhnavpiaor-r T=0.100 10-1 100 t101 102 T=0.250 signaledbyk 2forlargetoccursdifferentlyfordifferent Ir ≈ 10-1 100 101 t 102 103 104 N. In Fig. 5(c), we see that an isotropic liquid is not at all (b) VIr N=150 recoveredfor N = 75 up to the largest time of our simula- 2.0 Conf tion,evenathighT(=0.10).SystemwithN =500particles kIr (Fig. 5(a)) attains such limit more smoothly than N = 150 (Fig. 5(b)), indicating a surprisingly large temperature win- 1.0 2.0 dow of inhomogeneousliquid (as signaled by k < 2) will (b1) Ir 0.6 graduallyshrinkasournano-clusterapproachesthebulklimit. 1.0 kIr T=0.002 We find thatkIr attainsthe minimumvalue, which increases 0.3 T=0.006 withN, atT T . OurresultsforlowT indicatesthatthe X ∼ 10-1 100 t101 102 conjecture of universal exponential tails of Gs(r,t) in glass 10-1 100 101 t 102 103 104 formers[68]holdevenincaseofCoulombclusters. (c) VIr N=75 Similaranalysisforcircularconfinementshowsthatwhile 2.0 Conf k 1atlowT ,10.006. (c)t-dependenceofχ4(t)atdifferent T.χ4(t)attainsmaximumattimeτxwhichdecreaseswithincrease inT. (d) T dependence of τx along withτα, obtained fromQ(t). Signaturesof multi-scale temporalrelaxationsin confined BoththetimescalesshowsimilarT dependence. systemsisrecentlyelucidated[37]inourearlierstudy. Here, we focuson extractingrelevanttime scales for structuralre- laxationfromthe overlapfunction,cagecorrelationfunction byadistancegreaterthanr thenwesetw(r) = 0forallfu- c and persistence and exchange times. We will discuss how ture times. This definition ensures Q(t) is a monotonically these signify the underlying motional processes quantifying decayingfunctionoft. thenatureofthedynamics.Wewillfinallylookintothecross- Obviously,Q(t)dependsonthechoiceofr andtheopti- c correlationbetweenthesetimesscalestounderstandtheinter- malr ischosenasfollows:WestudythefluctuationofQ(t), c relationsbetweenthesetime-scales. definedasthedynamicalsusceptibility: 1 χ (t)= [ Q2(t) Q(t) 2] (12) 1. StructuralRelaxationTimefromOverlapFunction 4 N h i−h i at the lowest temperature for different choices of r . From We estimate the time evolution of the self part of a two- c definition, χ (t) = 0 when no particles move by a distance pointdensitycorrelationfunction,calledtheoverlapfunction 4 beyondr . Itiszero,aswell,whendisplacementsofallpar- Q(t)[69,70],definedas: c ticles are greater than r . So there must be a time scale, τ , c x whenχ (τ )reachesmaximumforagivenr .Thus,themax- 4 x c imumofχ ,i.e. thefluctuationsinQ(t),att=τ signifythe N 4 x 1 Q(t)= w(~r (t +t) ~r (t )) (11) maximaldynamicheterogeneity.AtagivenT,wechoosethe i 0 i 0 *N i=1 | − | + valueofrc forwhichsuchdynamicheterogeneityattainsthe X maximum. For example, we find that for N = 150, χ (t) 4 where w(r) = 1.0 if r < r and zero otherwise. The an- reaches its maximum for r = 0.09 which is about 14% of c c gular parentheses denotes averaging of results over the time r (seeFig.7(a)). AtagivenT,χ (t)attainsmaximumfora 0 4 origin,t ,andalsooverdifferentrealizationsofthedisorder. timeproportionaltothestructuralrelaxationtime[69,71]. 0 Since particles mostly rattle in a small distance aroundtheir Fig.7(b)showsthet-dependenceofQ(t)inirregularcon- equilibriumpositions(withinr ,asdefinedbelow)atsmallt, finementat differentT for N = 150. While at low T,Q(t) c Q(t) 1whereasforlarget,Q(t) 0signalingcompletion decays very slowly, it falls to zero rapidly for T > 0.006. ofstru∼cturalrelaxation. We evaluate∼w(r) bymonitoringthe The time dependence of Q(t) exp[ (t/τ )c], beyond a α ∝ − displacementofindividualparticlesandonceaparticlemoves veryshorttime,isfoundexponential(c = 1)forT > 0.006, 9 whereas,itisstretchedexponential(c 0.62)forT =0.006 1 0.7 and a much slower decay, seemingly≈power law for T = 0.8 (b) VICronf 0.6 0.6 τ 0.002. TT==00..000026 0.5 N=150 τNCC’ froTmo tgheet aonveerslatipmfautinocntioonf tQhe(ts)t,ruwcteuruaslereQla(xτaαt)ion=tim1/ee, τfoαr, C(t)g 0.4 TTT===000...000123000 τ 00..34 T 0.006. Forbulksystem,onecanestimatethestructural 0.2 T=0.050 prealra≥txoaftiothnetiimnteerbmyeldoioakteinsgcainttteoritnhgefteumncptoiornal,dFec(aky,to)f(thweitshelkf (a) VICronf N=150 00..12 s settothevalueatwhichthestaticstructurefactorfeaturesits 0.110-1 100 101t 102 103 104 0 0.01 0.02 T 0.03 0.04 0.05 firstpeak).But,sincethewavevectorkisnotwelldefinedfor afinitesystem,weuseQ(t)tocalculateτ . α FIG.8: (a)Decayofcagecorrelationfunction,cg(t),atdifferentT Next, we study the t-dependence of χ4(t) at different T forN = 150depictinghowtheinitialneighborsfail,withtime,to (Fig. 7(c)). We find that τx decreases to lower values with confineaparticle,ontheaverage.Thesolidityatlow-T isreflectedin increase in T. We show the T dependenceof τ along with thefactthatthesameneighborscontinuetocageaparticleatalltime. x ταinFig.7(d). TheT-dependenceofthesetwocharacteristic (b)T-dependenceofaveragecaging(τc)andnon-caging(τNC)time timesappeartobeidenticalwithinthetolerance.Thisimplies forN =150,illustratingthethermalcrossoveratT TX. ∼ that the enhanced heterogeneities cause structural relaxation in Coulomb clusters. The rapid increase of τ and τ with x α decreaseinT arereminiscentofglassysystems. Such decay is found for both the confinements (for circular confinement see Fig. 7 in supplementary material). Similar decayisalsocommontosupercooledliquids[73]. Recently, 2. Cage-correlationtime itisfoundthatwaterinnanoporoussilica[74]showsstretched exponentialdecaywiththecharacteristicsexponentβ = 0.6. Ref. [75] describingstretchedexponentialtemporaldecay in The dynamics in Coulomb clusters can be further probed supercooled liquids with β 0.6. While the literature dis- by addressing the ‘cage effects’. In glass formers, the par- ≈ cussesonlytheshort-rangeinteractingsystems,itisinterest- ticlesgetslower,withoutappreciablechangeinthestructure, ingto notethatourCoulombclusters(withlongrangeinter- uponapproachingtheglasstransitiontemperature.Thisisdue actions) show similar trends. For a system with with static tothecagingeffectinwhicheachparticleislockedupbyits random traps, the asymptotic value of β relates the dimen- Coulombrepellingneighbours. Rearrangementof ‘cage’re- sionalitydofthesystem throughβ = d/(d+2)[76]. Thus laxesthesystemandtherebytheparticlesdiffuse. Therateof ford= 2,asymptoticvalueofβ is0.5. Wefindthatforboth changeofsurroundingsofaparticleyieldsacagecorrelation thecircularandirregularconfinements0.45<β <0.6which function (CCF) and helps in understanding how rapidly the weconsiderisinbroadagreementwithsuchpredictions. localenvironmentofeachparticlechangesonanaverage. We estimate the cagingand non-cagingtimes by studying To calculate CCF we first define a neighbor list keeping the relative fluctuations in position of the particles with re- track of each particle’s neighbors. If the list of a particle’s specttoitsneighbors. Aperfectlycagedparticleisexpected neighborsattimetisidenticaltothelistattimet = 0,CCF to have equidistant neighbors. Exploiting this idea, we de- assumes unity for that particle. If, however, any of the confine the caging time (τ ) of the i-th particle as the time up fined particle’s neighborchanges, CCF becomeszero at that C instant of time. A neighborlist Li(t) for particle i in an N towhichthefollowingconditionholdsforatleastthreeofits neighboringparticles(j): particlesystemisavectoroflengthN,andisdefinedas[72] Li(t)=f(rij) (13) σi(t) d; foratleast j =3 ∆r (t) ≤ ij withj = 1,2,...,N,andf(r ) = 1ifj isthenearestneigh- ij bourof iat timet and zerootherwise. We use Voronoicon- Here j = 1,2, Nb, Nb denoting the number of near- ··· structiontoselectneighbourlistatallt. TheCCFattisgiven est neighbors of particle i; ∆rij(t) = ~ri(t) ~rj(t) and by σi2(t) = h∆2rij(t)i −h∆rij(t)i2. We t|ake cu−t-off va|lue d as0.1r (analogoustotheusualLindemannratio). Oncethe 0 Li(t) Li(0) abovecriterionbreaksdown, thenthe time takenby the par- C (t)= h · i (14) g L2i(0) ticletogetcagedagain,iscalledthenon-cagingtime(τNC). h i In above, caging is defined at least with respect to three of InFig.8(a)weshowthedecayofCCFatdifferentT forir- its neighbors, because in 2D a minimum of three particles regularconfinement.WeseethatatT =0.002,Cg(t)remains are required to ‘cage’ a particle. Fig. 8(b) shows the aver- closetounityforalltimes,implyingnosignificantrearrange- aged τ and τ for irregular confinements. Here, τ and C NC C ments of neighbours. This is because, particles are just jig- τ areexpressedrelativetothetotaltimeoftheMDsimu- NC glingonlyaroundtheirequilibriumpositions. AsT increases lation. Atlow T, cagingtime isexpectedlyhighasparticles Cg(t)showsadecaythatturnouttostretchedexponentialin aremostlyconfinedbyitsnearestneighbors. AsT increases t: c (t) exp[ t/τ ]β with β < 1 (details of the fitting thermalenergyovercomescaging.Thus,τ decreaseswithT g g C ∝ − parametersare given in Table I in supplementarymaterials). while τ increases. From Fig. 8(b) we see that these two NC 10 τP(log()) 000...456 (Na )= V50ICr0onf ττpe,, TTTTTT======000000......001200350522000000 (Nb=) 5V0CC0ronf ττep,, TTTTTT======000000......000012223505000000 Time scales111000123 (a)VCIronf <τpττ>αg (b)VCCornf <τpττ>αg 0.3 0.2 100 0.1 0 0.015 0.03 0.045 0 0.015 0.03 0.045 T T 0 -1 0 1 2 3log 4(τ) 5 6 7 8 -1 0 1 2 3log 4(τ) 5 6 7 8 9 103 (c)VCIronf τ2 τ1=τα, τ2=<τp> fFoIrG(.a9):irreDgiusltarribauntdion(b)ofciprecrusliasrtecnocnefin(τepm)eanntsdwexitchhNang=e (5τ0e0) tpiamrteis- 102 τ1=τα, τ2=τg cles. ThedecouplingofthetwodistributionsatlowT demonstrates 101 τ1= <τp>(×0.05), τ2=τg the qualitative similarityof the particledynamics in irregular traps withthoseinglassysystems. Suchdecouplingishardtodiscernfor 103 (d)VCCornf circulartraps. τ2 τ1=τα, τ2=<τp> 102 τ1=τα, τ2=τg τ =<τ >(×0.20), τ =τ timescalescrosseseachotheraroundthesameTX thatshows 101 1 p 2 g other signatures of the thermal crossover. We note that this 100 τ1 101 analysis excludes the particles on the boundary because the currentdefinitionof‘caging’isilldefinedforthem. FIG. 10: T-dependence of structural relaxation time τα, average persistence time τp and cage correlation timeτg for (a) irregular h i and(b)circularconfinements. Dottedlinesareforvisualguidance. 3. Persistenceandexchangetime Panel(c-d):Thickdotsshowthecross-Correlationbetweenτα, τp , h i and τg in a log-log plot for irregular and circular confinement, respectively,takenoveratemperaturerange0.01 T 0.05. Solid A crucialtime scale thatprobesthe relaxationmechanism ≤ ≤ linesrepresentthebestfittosuchcorrelations. τp isscaledbyap- as well as the glassiness of dynamics is the behavior of the h i propriatefactorsforvisualclarity. persistenceandexchangetimes[77,78]. Theyaredefinedas follows: At any given T, let us consider a particle i, whose initial(t=0)positionpositionisr (0). Thepersistencetime i thussuchdistributionlacksconfidencebelowT . X t for a givencut-offdistance d is specified by the first time 1 Since, the overlap function, cage correlation function and thatparticleihasmovedfarenoughthat r (t ) r (0) r . i 1 i p persistence time, all are connected with the structural relax- | − |≥ ationofthesystem,wenowlookintotheT-dependenceofthe Exchangetimest t forn> 1,requiretherecursive n n−1 characteristictimescalesassociatedwiththesequantities.For − determinationt ,settingt asinitialtime. Foragivenr , n n−1 p example,τ (obtainedbyfittingthecagecorrelationfunction g distributionsofexchange(τ )andpersistence(τ )timescan e p c (t)asdiscussed)representscharacteristictimescaleforthe g beobtainedbyensembleaveragingoveralltrajectories. r is p rearrangementofparticlesintheirlocalenvironmentandaver- chosen of the order of r so as to probe the structuralrelax- 0 agepersistencetime, τ ,depictsthetimerequiredforparti- p ationanddiffusion.Thesamer isusedforallT.Ithasbeen h i p clestomoveacertaindistanceonaverageandthusassociated shown[77, 78] thatfor glassy dynamics,distributionof (τ ) e withthe relaxationmechanismin thesystem. InFig.10, we and(τ )generallydecoupleneartheglasstransitiontemper- p showtheT-dependenceofτ ,τ and τ forirregular(panel α g p ature. Thisdecouplingisattributedtodynamicheterogeneity h i a)andcircularconfinements(panelb). We findthatallthese in thesesystems, andconstitutea keyfeatureofsupercooled time scales increase rapidly for IWM compared to CWM as liquids[77,78]. T approachesT . Thus, the signaturesof glassy dynamics, X ThedistributionofthesetwotimescalesareshowninFig.9 whileshowupinbothconfinements,aremoreprominentfor forirregular(panela)andcircular(panelb)confinementswith irregularconfinement,broadlymimickingabulkdisordersys- N = 500. We notice that for T > 0.030, P(τ ) and P(τ ) e p tem. coincide for both the confinements. As T becomes close to T ( 0.020), the two distributions become distinct; mean X ≈ valueforP(τ )movestowardlongertimesforIWM.Incon- p trast, for circular confinementdistribution of these two time scales do not appear to decouple. We have ensured that the 4. Cross-Correlationbetweendifferentrelaxationtimes decouplingisnotbecauseofbiasinthesamplingprocessby carryingoutJackknifetest[79]. Suchdecouplingisalso ob- Here, we studythe cross-correlationbetweenthe different servedforN = 150(seeFig. 8insupplementarymaterials). time scales described above. Fig. 10(c) illustrates the cross- Notethatthedistributionsofτ andτ areshownforT T correlationbetweenτ andτ ,τ and τ , and τ andτ , e p X α g α p p g ≥ h i h i as at low T, displacementof particles beyondr is rare and gatheredoveratemperaturewindowacrossmelting;namely, p

Static and Dynamic Properties of Two Dimensional Coulomb Clusters

Similar Static and Dynamic Properties of Two Dimensional Coulomb Clusters