The mechanism of the drastic slowing down of the structural

Transcription

1 Frustrtion on the wy to crystlliztion in glss HIROSHI SHINANI AND HAJIME ANAKA* Institute of Industril Science, University of okyo, --1 Kom, Meguro-ku, okyo , Jpn *e-mil: Pulished online: 19 Ferury ; doi:1.13/nphys35 Some liquids do not crystllize elow the melting point, ut insted enter into supercooled stte nd on cooling eventully ecome glss t the glss-trnsition temperture. During this process, the liquid dynmics not only drsticlly slow down, ut lso ecome progressively more heterogeneous. he reltionship etween the kinetic slowing down nd growing dynmic heterogeneity is key prolem of the liquid glss trnsition. Here, we study this prolem y using liquid model, with crystlline ground stte, for which we cn systemticlly control frustrtion ginst crystlliztion. We found tht slow regions hving high degree of crystlline order emerge elow the melting point, nd their chrcteristic size nd lifetime increse steeply on cooling. hese crystlline regions led to dynmic heterogeneity, suggesting connection to the complex free-energy lndscpe nd the resulting slow dynmics. hese findings point towrds n intrinsic link etween the glss trnsition nd crystlliztion. he mechnism of the drstic slowing down of the structurl relxtion of liquid on cooling is one of the centrl issues of the physics of the liquid glss trnsition. In reltion to this, experimentl 1 nd numericl 5, evidence hs een ccumulted for the existence of dynmic heterogeneity in supercooled stte of liquid, whose length scle increses on cooling. he link etween this growing length scle nd the slowing down of the structurl relxtion hs een ctively discussed on the sis of the Adm Gis theory 7, the spin glss model,9, the mode-coupling theory 1 nd the kineticlly constrined lttice model 11,sitmyprovide clue to our understnding of the liquid glss trnsition. Despite intensive efforts, however, the physicl fctors tht control dynmic heterogeneity remin elusive. Mny different ides hve een proposed on the origin of the liquid glss trnsition 1 1. Among them, frustrtion hs often een considered to ply key role in the glss trnsition in conjunction with other glssy systems such s spin glss nd dipolr glss 1. he most populr scenrio is geometricl frustrtion ssocited with the fct tht loclly fvoured structures such s icoshedrl order cnnot fill up the spce 1 1. In other words, liquids tend towrds glol icoshedrl order, ut they cnnot chieve it, whichistheoriginoffrustrtioninthisscenrio.hereislso different possiility 17,1 (seeref.19onthedifference in the physicl mechnism etween the two scenrios): liquids tend to order into the equilirium crystl, ut frustrtion effects of loclly fvoured short-rnge ordering on long-rnge crystlline ordering prevent crystlliztion nd help vitrifiction. Even simple one-component liquid my suffer from such frustrtion if the locl symmetry of the interctionpotentildoesnotperfectlymtchthesymmetryofthe equilirium crystl. Motivtedythisfrustrtionscenrio,wehvedeveloped different type of simultion model, where we cn systemticlly chnge the degree of frustrtion ginst crystlliztion. Moleculr dynmics simultion is useful mens for investigting the dynmics of liquids microscopiclly. We cn lso directly control the intermoleculr potentil in moleculr dynmics, wheres it is difficult to do so in experiments. We modify sphericlly symmetric interction potentil y including n nisotropic prt; morespecificlly,weintroducespindirectoru i into prticle i such tht the interction etween prticle i nd j depends not only on the interprticle distnce r ij, ut lso on the ngles mong nturephysics VOL MARCH Nture Pulishing Group

2 Δ =. Δ =. Δ =.5 Δ = Δ =. Δ = Liquid V 1. U 3.5. Plstic crystl Supercooled liquid Glss c U Δ =.5 Δ =. 3. AFM crystl.1 Plstic crystl d U AFM crystl Liquid Figure 1 Phse chnge on cooling nd heting.,, -dependence of V () nd U () for vrious Δ on cooling. he rrow indictes the loction of g.for Δ.5, we see step in the volume chnge, reflecting crystlliztion into plstic crystl. For Δ =., on the other hnd, there is no step nd only the slope of d V /d chnges t g. c,d, -dependence of U on heting re shown for Δ =.5(c)ndΔ =.(d). For Δ =.5, we see two steps, reflecting the AFM-to-plstic crystl trnsformtion nd the melting of the plstic crystl. For Δ =., on the other hnd, we see only one step, which reflects the melting of the AFM crystl. u i, u j,ndr ij (see the Methods section). he modultion strength is given y prmeter Δ. his potentil loclly fvours fivefold symmetry (see Supplementry Informtion, Fig. S1), ut hs crystlline stte with long-rnge ntiferromgnetic (AFM) spin order s the ground stte. As five-fold symmetry is not consistent with ny crystllogrphic symmetry 13,1,thisΔ cn e regrded s the strength of frustrtion ginst crystlliztion. o the est of our knowledge, this is the first two-dimensionl glss-forming model liquid system tht consists of single-component prticles nd hs crystlline stte s the ground stte. By using this lttice-free spinglss model, we investigte the roles of frustrtion in vitrifiction s well s the reltionship etween crystlliztion nd vitrifiction. We investigted the phse ehviour of this system s function of Δ y studying the temperture () dependence of the verge volume per prticle V nd the verge potentil energy per prticle U t pressure P =.5 forfixedcoolingrteq = 1 (see Fig. 1,). All results re given in reduced units (see Methods). For wek frustrtion (tht is, for Δ <.), the system crystllizes (intoplsticcrystl)oncoolingwithdistinctjumpinthe volume. On the other hnd, for strong frustrtion (tht is, for Δ.), there is no jump in the volume itself, ut there re rther shrp chnges in oth d V /d nd d U /d round the glss-trnsition temperture ( g ). his indictes tht the system is vitrified without crystlliztion under strong frustrtion even for the sme cooling rte Q: energetic frustrtion ginst AFM crystl Figure he phse digrm of our model system. he horizontl xis represents Δ, which is mesure of the strength of frustrtion ginst crystlliztion. he filled circles represent the melting point of plstic crystl. he open squres represent the temperture where n AFM crystl trnsforms into plstic crystl or melts on heting. he filled squres represent the g, which is defined s τ α ( g ) = 1. he filled tringles represents the Vogel Fulcher temperture. =.99,.7 nd. for Δ =.,.7 nd., respectively. he increse in g with Δ mens tht liquid ecomes stronger (tht is, less frgile) with n increse in Δ (see the text). We lso show simulted structure corresponding to ech stte of our system. he rrows indicte the spin direction. crystlliztion prevents crystl nucletion. o understnd wht kinds of crystlline ordering tke plce in this system, we studied crystllinestructuresthtcneformedtp =.5 for vrious Δ. We found tht the lowest-energy ground stte of our system t Δ > is crystl with AFM spin order (see Fig. 1c). his crystl hs trnsltionl symmetry concerning oth position nd orienttion of prticles. Note tht this crystlline lttice is not perfectly hexgonl, ut slightly distorted unixilly ecuse of strong nisotropic interctions. We lso studied the chnge of the potentil energy on heting to work out the phse ehviour for vrious vlues of Δ (see Fig. 1c,d). For wek frustrtion (for Δ <.),thecrystlwithafmspinorderfirsttrnsformsinto plstic crystl to gin the rottionl entropy, nd then the plstic crystl melts into liquid t m (Δ), where m is the melting point. his plstic crystl hs hexgonl positionl order ut no orienttionl order. For strong frustrtion (for Δ.), onthe other hnd, no plstic crystl exists nd the crystl with AFM spin order melts directly into liquid. Interestingly, in the region where, on heting, the AFM crystl directly melts into liquid without pssing through the plstic crystl (nmely, for Δ.), liquid vitrifies without crystlliztion on cooling. Figure summrizes the phse ehviour of this system including the non-equilirium stte sfunctionofthedegreeoffrustrtionδ. We clculted the -dependence of the rdil distriution function g(r). Figure 3 shows such n exmple for Δ =.. here is no long-rnge correltion even t low tempertures, which indictes tht the system forms glssy stte for this vlue of Δ. hus, we confirm tht liquid vitrifies on cooling without crystlliztion for Δ.. As explined in the Methods section, our system tends to form loclly fvoured structures of five-fold symmetry. From the -dependence of g(r), we cn clerly see tht oth clusters with crystllogrphic symmetry nd loclly fvoured structures progressively develop with decrese in. We confirmed tht the -dependence of the height of pek corresponding to the Δ nturephysics VOL MARCH 1 Nture Pulishing Group

3 Log g (r ) τ α r c 1. * m * m.7 β / 1/ g (r ) r 1 15 Figure 3 Chnge in structures nd dynmics on cooling., -dependence of the rdil distriution function g(r )forδ =.. is.1,.,.5 nd. from the top to the ottom. g(r )inwiderr rnge for =.1 is shown in the inset. he rrows t r 1.13 nd r 1.3 correspond to pek of the AFM crystl nd pek of the loclly fvoured structure of five-fold symmetry, respectively (see the lck lines in the digrms of the two structures). his cn lso e confirmed in rel spce in Fig.., -dependence of τ α for Δ =.. he dshed line is the Arrhenius -dependence fitted to the dt ove m, wheres the solid line is the Vogel Fulcher ehviour fitted to ll of the dt. c, -dependence of β for Δ =.. Owing to strong nisotropic interctions, the relxtion function is not single exponentil (β.95) even t high tempertures. Such ehviour is known for hydrogen-onding liquids such s glycerol. loclly fvoured structure is consistent with the prediction of our two-order-prmeter model of liquid (see eqution () in ref. ) tht ssumes tht liquid consists of only two types of structure; nmely, loclly fvoured structures (lue prticles in Fig. ) nd norml-liquid structures (green prticles in Fig. ). We lso clculted the rottionl utocorreltion function C R (t) = (1/N) u i i(t) u i (), wheret is time nd N is the numer of prticles. hen we fitted the stretched exponentil function to C R (t) s C R (t) exp( (t/τ α ) β ).Figure3ndc shows the -dependence of the chrcteristic rottionl relxtion time τ α nd the Kohlusch exponent β otined y the fitting, respectively, for Δ =.. τ α cn e well-fitted y the Vogel Fulcher lw τ α = τ exp(d /( )) with τ =.1, D = 7. nd the Vogel Fulcher temperture =.99. he increse of τ α on cooling is often clssified etween strong nd frgile extremes using g s scling prmeter 3,5. Liquids whose τ α oeys the Arrhenius lw re clled strong, wheres frgile liquids hve super-arrhenius ehviour. D is n indictor of the strength of liquid: the lrger D is, the stronger the liquid is. Here D = 7. corresponds to frgile liquid. he -dependence of τ α strts to devite from the Arrhenius ehviour elow m (=.), which we define s m of crystl free from frustrtion (tht is, t Δ = ) 17. β is lmost constnt ove m, wheres it monotoniclly decreses with decrese in elow m. Both results indicte tht liquid shows complex coopertive ehviour only elow.welso m clculted the men squre displcement r (t) (detiled lter) nd the intermedite scttering function nd found tht there is two-step relxtion (fst β nd α relxtion) t tempertures lower thn.weconfirmeddynmicheterogeneityofsupercooled m liquid y mens of time-dependent four-point density correltion function (see the Supplementry Informtion) nd found tht dynmic heterogeneity ecomes more pronounced with decrese in (weprovidefurtherdetilslter).hus,wemysythtthis system reproduces the most essentil fetures of the liquid glss trnsition well, which re known from previous studies 1. Here we focus on the reltionship etween the dynmics nd the structure; specificlly, the medium-rnge crystlline order. It ws pointed out 1,1 tht reveling the reltionship etween dynmics nd structure is key to the understnding of the slow dynmics ssocited with the liquid glss trnsition. his prolem is lso relted to the prolem of the complex energy lndscpe 5,. he most populr method is to focus on n inherent structure 5 or n idel glss structure, which is elieved to form locl minim (sins) in the energy lndscpe. In contrst to this populr model, herefter we propose possiility tht crystlline structure is key structure of supercooled stte. First, we show evidence for the existence of long-lived clusters of medium-rnge crystlline order in supercooled liquid stte. In our system, we know the equilirium crystlline structures (see Fig. ). Note tht only one type of crystl, which hs AFM spin order, exists in the glss-forming region (Δ.). he order prmeter Ψ i chrcterizing it cn then e defined s Ψ i (t) = (1/) u i,j i(t) u j (t), where i, j represents the summtion over the nerest neighours of prticle i.whenψ i., we judge tht prticle i is involved in clusters of high crystlline order. We judge tht prticle j is lso involved in the sme cluster if the spin of prticle j is prllel or ntiprllel to tht of prticle i; specificlly, if u i u j.9. o void the effects of therml noise nd detect regions with slow dynmics, we verged the order prmeter over the durtion of τ α s Ψ i (t) = (1/τ α ) t +τ α / t τ α / dt Ψ i (t ). We emphsize tht this orienttionl order prmeter is much more sensitive to the crystlline order thn the positionl order prmeter, which is one of the merits of our model system. In Fig. c, we present snpshots of the sptil distriution of the orienttionl order prmeter for three tempertures. At =.35, there re few clusters of high crystlline order. With decrese in, clusters of high crystlline order pper nd grow in size nd lifetime. We lso present rottionl dynmics in Fig. d f nd trnsltionl dynmics (prticle trjectories) in Fig. g i corresponding to Fig. c. Despite the fct tht Fig. c re snpshots, Fig. d f show the timeverged informtion nd Fig. g i show trjectories, they re well-correlted: prticles re less moile oth rottionlly nd trnsltionlly in ordered regions thn in disordered regions. o demonstrte this correltion more quntittively, we lso clculted the rottionl correltion function C R (Ψ, t) nd the men squre displcement r (Ψ, t) for regions hving certin rnge of the orienttionl order prmeter (Ψ i Ψ) t t =. Here C R (Ψ, t) = (1/N Ψ ) Ψ i Ψ u i (t) u i (), wheren Ψ is the numer of prticles elonging to Ψ i Ψ nd r (Ψ,t) = (1/N Ψ ) [r Ψ i Ψ i(t) r i ()].Asshown in Fig. 5, we revel tht the higher the degree of crystlline order, the slower the locl rottionl relxtion. he correltion etween trnsltionl motion nd the degree of the crystlline order cn lso nturephysics VOL MARCH Nture Pulishing Group

4 c d e f g h i Figure Medium-rnge crystlline order nd dynmic heterogeneity. c, Snpshots of liquid structures (t certin time t s ) for three different tempertures:, =.35,, =.1 nd c, =.17 (see lso Supplementry Informtion, Video S1, which corresponds to ). Red prticles hveψ i greter thn.75. Blue prticles, on the other hnd, represent loclly fvoured structures with five-fold symmetry. he rrows indicte the spin direction or the direction of u i. d f, he slowness of the rottionl dynmics. he red nd lue prticles denote those slowly rotting during t s < t < t s + τ α /: the prticles elonging to the loclly fvoured structures re coloured lue. g i, he trjectories of corresponding to c. he trjectories in g i re drwn from t s to t s + τ α, t s +τ α nd t s +τ α, respectively. Note tht c re snpshots t t s. eseeninfig.5.infig.5cnddwelsoshowthedistriutionsof C R (Ψ,t 1 ) (t 1 = 1,) nd r (Ψ,t ) (t = 7,9), respectively, where t 1 roughly corresponds to the rottionl relxtion time, τ α (more precisely, t 1 =.7τ α ), nd t is the time when the heterogeneity of the trnsltionl motion, which is chrcterized y χ (see the Supplementry Informtion), ecomes mximum. hese results clerly indicte positive correltion etween the degree of crystlline order nd the slowness of the structurl relxtion, tht is, link etween crystlline order nd dynmic heterogeneity. he ove results show tht clusters of high crystlline order emerge elow m nd their verge size nd lifetime increse on cooling. he lifetime is longer thn τ α (see Supplementry Informtion, Fig. S) nd reches the order of 1τ α ner g.his is consistent with experimentl findings. Prticles do not rotte for long time in these clusters, s cn e seen from Supplementry Informtion, Video S1. As this heterogeneity hs finite lifetime, it cn e regrded s dynmic heterogeneity. he clusters re temporlly creted nd nnihilted, nd their verge size remins constnt with time. hus, we speculte tht the size of the clusters is still smller thn the size of the criticl crystl nucleus, ut this needstoecheckedcrefullyinthefuture. Figure shows the -dependence of the chrcteristic size of clusters, ξ. We define this size ξ() s ξ() = N,where N is the verge numer of prticles contined in cluster with Ψ.75. We lso plot the dynmic coherence length otined y the nlysis using the four-point time-correltion function, ξ (see the Supplementry Informtion), in Fig.. Note tht ξ chrcterizes the heterogeneity of trnsltionl motion (see lso Fig. g i). We found tht ξ is proportionl to ξ, nd oth ξ nd ξ cn e fitted y function of ξ () () = A () ( ) /d (d is the sptil dimensionlity; d = in our cse) with =.99, which is otined y the fitting of the Vogel Fulcher reltion to the -dependence of τ α (seefig.3). he ehviour of ξ nd ξ ove is consistent with the scling rgument,9, which connects the increse of ξ nd ξ to tht of τ α. nturephysics VOL MARCH 3 Nture Pulishing Group

5 otl C R ( Ψ,t )... otl Ψ =.5 Ψ =.7 Ψ =.75 Δr ( Ψ,t ) Ψ =.5 Ψ =.7 Ψ =.75 Ψ =. Ψ =.5. Ψ =. Ψ =.5 1 c Log t Log t d Frequency otl Ψ =. (LFS) Ψ =.5 Ψ =.7 Ψ =.75 Ψ =. Ψ =.5 Slow Frequency Slow otl Ψ =. (LFS) Ψ =.5 Ψ =.7 Ψ =.75 Ψ =. Ψ = C R ( Ψ,t 1 ) Δr ( Ψ,t ) Figure 5 Reltion of rottionl nd trnsltionl dynmics to the degree of locl crystlline order., C R (Ψ, t ) for =.17 nd Δ =.. Note tht the dynmic heterogeneity smers out for lrge t., r (Ψ, t ) for =.17 nd Δ =.. c, he distriution of C R (Ψ, t 1 )(t 1 = 1,). HereΨ vlues re ±.. he prticles with lrgerψ re slower in rottion, ut the prticles elonging to the loclly fvoured structures (LFS,Ψ.) re lso slow. d, he distriution of r (Ψ, t )(t = 7,9). he prticles with lrgerψ re slower in trnsltion. Unlike rottionl motion, the prticles elonging to the loclly fvoured structures re rther fst in trnsltionl motion. Note tht these distriution functions re normlized such tht the integrl ecomes one. In ddition to the growth of clusters, the increse in the frction of ordered clusters nd the resulting percoltion nd/or the increse in the frictionl contcts etween clusters my lso cuse slow dynmics. Although we cnnot specify which is the most relevnt mechnism for slow dynmics, we my sy tht dynmic heterogeneity nd the slow dynmics re intrinsiclly relted to medium-rnge crystlline ordering in supercooled liquid. Next we focus on nother importnt structure, tht is, loclly fvoured structure, which induces frustrtion ginst crystlliztion. We found tht the rottionl dynmics of prticles in loclly fvoured structures is slow (see Figs d f nd 5c), lthough their trnsltionl dynmics re not so slow (see Fig. 5d). hus, oth medium-rnge crystlline ordering nd short-rnge ond ordering contriute to slow rottionl dynmics ner g. However, s the sptil distriution of loclly fvoured structures is rther rndom, their connection to dynmic heterogeneity (or the growing length scle) remins the future prolem. We note tht medium-rnge crystlline ordering cn lso e confirmed in the rdil distriution function g(r) shown in Fig.3. We cn clerly see tht the height of the pek corresponding to the crystlline order increses with decresing. We do not find ny significnt chnge in the structure fctor S(q) (q is the wvenumer)inthelow-q region (q 1/ξ). his is consistent with the well-known fct tht dynmic heterogeneity cnnot e detected in smll-ngle scttering experiments. he reson my e due to thefctthtclusterswithhighcrystllineorderpperrndomlyin spce without sptil correltion. Our simultion results indicte tht there is possiility to detect dynmic heterogeneity, or medium-rnge ordering, in g(r) nd S(q) not in low-q region where q 1/ξ, ut rther in high-q region where q 1/σ (σ is the prticle dimeter) through medium-rnge crystlline ordering. In reltion to this, it is worth mentioning the experimentl study 7 on the -dependence of S(q) in propylene cronte nd phenyl slicilte (slol). here it ws found tht shoulders in S(q), whichwereidentifiedspeksfromclustersofhigh crystlline order, progressively develop in the supercooled liquids elow m on cooling. Our simultion results re consistent with this experimentl finding, which my imply tht this ehviour is generic to ny glss-forming liquids. Further experimentl studies long this line re highly desirle. Our study indictes tht glssy slow dynmics cn e induced y frustrtion etween long-rnge density ordering towrds crystlliztion nd short-rnge ond-ordering towrds the formtion of loclly fvoured structures. According to this 17 19, liquid suffering from stronger frustrtion should e stronger. In our simultions, we found tht D increses with n increse in Δ, s shown in Fig. : D = 7., 1.9, 17., 9.7 nd.3 for Δ =.,.5,.7,.75 nd., respectively. We stress tht this rnge of D nturephysics VOL MARCH Nture Pulishing Group

6 ( ) ξ ξ ξ Δ =. 5 Δ =.5 Δ =.7 Δ =.75 Δ =. 3 1 τ α Log g / θ i is n ngle etween the reltive vector r ji = r j r i nd the unit vector u i, which represents the orienttion of the xis of prticle i. θ j is n ngle etween the reltive vector r ij = r i r j nd the unit vector u j (see Supplementry Informtion, Fig. S1). he function h((θ θ )/θ c ) (θ = 1 nd θ c = 53.1, where θ c is the width of the ngulr prt of the potentil) hs mximum t θ = 1, nd θ is n energeticlly fvoured vlue of θ i, nd is the ngle where h((θ θ )/θ c ) ecomes mximum. hus, this term stilizes the loclly fvoured structure of five-fold symmetry, s shown in Supplementry Informtion, Fig. S1. Δ controls the stility of the loclly fvoured structure s well s the strength of frustrtion ginst crystlliztion. Our system hs frustrtion tht is relted not only to the orienttion of the prticle xis, which my e regrded s spin, ut lso to the position of the prticle. In this sense, our model cn e considered s lttice-free spin system with trnsltionl degrees of freedom (lttice-free spin glss). Figure Growth of dynmic heterogeneity on cooling nd the reltionship etween Δ nd the frgility., -dependence of the chrcteristic size of crystl-like clusters ξ nd the correltion length of trnsltionl motion ξ otined from the nlysis of the four-point time-correltion function (see the Supplementry Informtion) for Δ =.. he solid nd dshed lines re the fittings of A/( ) with =.99 for ξ nd ξ, respectively. We found tht ξ ξ. he error r indicte the rnge of ξ oserved during 1τ α., he Angell plot of τ α ginst g /. g =.15,.15,.13,.17 nd.1 for Δ =.,.5,.7,.75 nd., respectively. lmost covers from the frgile (for exmple, o-terphenyl, D 5) to the strong extremes (for exmple, silic, D ). his is nother indiction tht the frustrtion ginst crystlliztion controls the nture of the liquid glss trnsition including the frgility of liquid. Network-forming tendency seems not to e prerequisite for mking liquids strong, in contrst to common eliefs. Our study suggests tht the model where the slow dynmics re cused y jmming owing to dense pcking my not e enough to understnd theglsstrnsition.itmyeimportnttoconsiderwhysystem cn remin in disordered stte nd e jmmed without ordering; nmely, frustrtion dversely ffects crystlliztion. MEHODS MODEL Here we use moleculr dynmics simultion to investigte the roles of crystlliztion in glss trnsition in the spirit of our two-order-prmeter model of the liquid glss trnsition For this purpose, we introduce potentil tht cn directly nd systemticlly control the strength of frustrtion ginst crystlliztion. U(r ij,ω ij ) = U(r ij ) + U(r ij,ω ij ), where r ij is the distnce etween two prticles i nd j, nd Ω ij expresses the orienttion etween these prticles. he first term of the potentil is n isotropic potentil tht tends to mximize the density of liquid nd encourge crystlliztion. he second term is n nisotropic potentil tht leds to the formtion of loclly fvoured structures. In this pper, we dopted the Lennrd Jones potentil s the isotropic potentil U(r ij ): U(r ij ) = ɛ ( ( σ r ij ) 1 ( ) ) σ, r ij where ɛ is the well depth of the potentil nd σ is the dimeter of the prticle. he nisotropic prt of the potentil is given y ( ) σ [ U(r ij,ω ij ) = ɛ h r ij ( θi θ θ c ) + h ( θj θ θ c ) ] 35π θ c, h(x) = 1 3x +3x x ( 1 < x < 1); h(x) = (x 1, x 1). SIMULAION MEHOD he numer of prticles in our simultions is N = 1,. We used periodic oundry conditions. We cut off nd smoothed oth the potentil nd its derivtive t 3.σ to sve computtionl time. Here we show ll results in reduced units. We used prticle mss m, σ nd ɛ s the sic unit of mss, length nd energy, respectively. hus, the moment of inerti I, the temperture, the pressure P, the distnce r nd the time t re scled s I I/mσ, k B /ɛ (k B is Boltzmnn s constnt), P Pσ /ɛ, r r/σ nd t t/τ (τ = mσ /ɛ). o control nd P of system, we used the Nosé Poincré Andersen (NPA) thermostt nd extended the NPA method to include the rottionl degrees of freedom of prticles. Our modified NPA hmiltonin (H NPA )is given y H NPA (t) = (H NA (t) H NA ())s, H NA = p mvs + p φ Is + U(qV 1/,{φ i }) + gk B lns + Π V + Π s + PV, M V M s wherenaisthenosé Anderson hmiltonin, p is the momentum, p φ is the ngulr momentum, φ i is the ngle etween x xis nd u i, g is the totl numer of the degrees of freedom of this system, s is the vrile to control temperture, V is the two-dimensionl volume of the system, Π V is the momentum conjugte to V, Π s is the momentum conjugte to s, M V is the mss of V, M s is the mss of s nd q is scled position vector. Here we used I =., M V =.1 nd M s = 1.. We confirmed the equiprtition of the energy into the trnsltionl nd the rottionl degrees of freedom of prticles. According to the theoreticl estimte, the vrince of the internl temperture distriution should e /N nd /N for trnsltion nd rottion, respectively, which is lso confirmed in our simultions within n ccurcy of %. We lso proved tht the ove NPA method extended to system with the rottionl degrees of freedom constitutes the NP ensemle. he results shown in this rticle re for P =.5 nd cooling rte of Q d/dt = 1.A system with the Lennrd Jones potentil U(r) melts t m =. t P =.5. After quenching the system into trget temperture, we nneled it for long time (during 3 τ α ) efore mking ny nlysis. Received 1 Septemer 5; ccepted 1 Jnury ; pulished 19 Ferury. References 1. Sillescu,H.Heterogeneity t the glss trnsition: review.j. Non-Cryst. Solids 3, 1 1 (1999).. Ediger, M. D. Sptilly heterogeneous dynmics in supercooled liquids. 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