Model for dynamics of structural glasses

PHYSICAL REVIEW E MARCH 1997VOLUME 55, NUMBER 3

Model for dynamics of structural glasses

Ian S. Graham,1 Luc Piche,2 and Martin Grant31Instructional and Research Computing, Computing and Communications, 4 Bancroft Avenue, University of Toronto,

Toronto, Ontario, Canada M5S 1A12National Research Council of Canada, Industrial Materials Institute, 75 De Mortagne Boulevard,

Boucherville, Que´bec, Canada J4B 6Y43Physics Department, McGill University, Rutherford Building, 3600 rue University, Montre´al, Quebec, Canada H3A 2T8

~Received 7 July 1995; revised manuscript received 28 May 1996!

We have studied, using dynamical Monte Carlo methods, a facilitated kinetic Ising model for structuralglasses. We observe stretched-exponential decays (a relaxation! of the equilibrium spin autocorrelation func-tion at late times and are able to estimate accurately the corresponding relaxation timest. These are found todiverge at nonzero temperatures following a Vogel-Fulcher law. In addition, we observe early-time exponentialrelaxation analogous tob relaxation of glass formers. We also examined the effective thermodynamics ofsystems quenched below this divergent temperature and subsequently heated. The result is a peak in thespecific heat with properties matching those of the putative glass transition. We also find that these peaks, fordifferent heating rates, can be rescaled to lie on a universal curve. Finally, we studied the evolving structurefactor of a frozen glass following a rapid jump in temperature. The results are qualitatively the same as thosefor x-ray studies of heated glasses. We thus demonstrate that this purely dynamical microscopic model canreproduce much of the phenomenology of the glass transition and near-glass relaxation, with no tunableparameters.@S1063-651X~97!07301-7#

PACS number~s!: 05.50.1q, 43.20.1g, 63.20.Pw, 71.55.Jv

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I. INTRODUCTION

Recently, there has been renewed experimental and tretical interest in structural glasses and the glass transiNew experimental methods have allowed for detailed stof the dynamical processes taking place in highly viscofluids. At the same time new theoretical approaches, in pticular mode-coupling theories, have provided explanatifor some of these observations, in particular in the mediuto high-viscosity region above the experimental glass tration. However, the nature of the highly viscous fluid nearglass, and the glass transition itself, remains unreso@1–4#.

Unlike a spin glass, the Hamiltonian of glass-forming mterials contains no intrinsic disorder. Instead the structudisorder arises from a dynamical mechanism~rapid cooling,for example! that keeps the system from forming an orderequilibrium phase. It is currently believed that the resultistructural glass may be not a proper equilibrium phaseinstead a material trapped in a long-lived metastable stHowever, the dynamical mechanism causing this trappinunclear, as is how such a mechanism can lead to theserved experimental phenomenology. For example, givemechanism we would like to know how its onset, as oapproaches the glass transition temperatureTg , gives rise tononexponential decays of the autocorrelation or relaxafunctions and to an apparent divergence in the relaxatime at a nonzero temperatureT0 belowTg , often character-ized by the Vogel-Fulcher lawt;eA/k(T2T0)@5#, whereA is aconstant energy scale andk is Boltzmann’s constant. Wealso would like to know how the mechanism can yield thmal properties of the glass transition that mimic closely thof a true thermodynamic phase transition. Finally, we wolike to know if the transition to a glass represents a new c

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of true dynamical transitions. Given these questions, meffort has been devoted to developing theories and mofor glasses and glassy relaxation. To date, however, theno satisfactory description relating a microscopic mechanfor slowed relaxation to the relaxation time divergence nthe glass transition or to the glass transition itself. Findsuch a mechanism would be of great use in the developmof a rigorous theory for glasses.

In this paper we test the idea that dynamical frustrationthe relevant mechanism. We do this using a microscopicnamical spin model for glasses and glassy relaxation.starting point is a kinetic Ising model in theabsenceof spin-spin interactions (J50). By itself this model is trivial: withJ50 there is no thermodynamic transition at any tempeture. All relaxation and correlation functions decay exponetially, in or out of equilibrium, and all spins are uncorrelateWe introduce dynamical frustration by adding a dynamirule for frustration to the Monte Carlo dynamics. This rudefined below, satisfies detailed balance, so that the evenequilibrium state remains unchanged, but the dynamical pcess leading to equilibrium changes markedly. Such an iwas proposed in the facilitated kinetic Ising model of Frdrickson and Andersen@6#. Their analytic study predictedthat their model should have a power-law singularity in tequilibrium relaxation time at nonzero temperature, whthey interpreted as a dynamical transition leading to a glaalthough subsequent simulation studies did not supportexistence of this power-law singularity@7,8#.

Notwithstanding the absence of the power-law singulity, we have revisited this model and made a large-scnumerical study. This is because we believe its dynamfrustration can provide the most simple representationglassy behavior. We have used a simplified version offacilitation model, which we studied in both two and thre

2132 © 1997 The American Physical Society

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55 2133MODEL FOR DYNAMICS OF STRUCTURAL GLASSES

dimensions and for several facilitation rules. We examinthe properties of the model for temperatures where equrium was possible and also the effective thermodynamupon heating from and cooling into the essentially glastate. Our results are as follows. At low temperaturesequilibrium autocorrelation functions have the late-timstretched exponential decays characteristic of glass-formmaterials. More importantly, the relaxation timet that weextract from these data diverges at nonzero temperatureslowing a Vogel-Fulcher law. In addition, our simulationsheating and cooling of a quenched low-temperature ‘‘glasshow a peak in the effective specific heat with propertvery similar to those of the glass transition. We also find tdata for different heating rates can be scaled onto a univecurve. These results are independent of the dimensionalithe model or of the facilitation rules employed. Finally, ostructural study of rapidly heated glasses shows behasimilar to that seen in dynamical x-ray studies of rapidheated glasses. Thus this simple model is able to reprodmuch of the phenomenology of glassy relaxation andglass transition, with no adjustable parameters@9#.

The paper is organized as follows. In the next sectionreview the experimental phenomenology of glasses alwith the results of molecular-dynamics simulations of mosystems. Following this is a review of some models atheories. We next introduce our model, discussing its impmentation and theoretical underpinnings. Finally, we presour results and discuss them in the context of current expmental measurements on glasses.

II. REVIEW

Glasses have been extensively studied over the pastcentury, so that the basic phenomenology is well knownhas been extensively reviewed@1–4#. Here we summarizethese general results. We discuss in more detail experimresults on dynamics in glassy materials, as well as the stof current theory.

A qualitative understanding of the glass transition canobtained from the free-volume approach@10,11#. This modelpredicts a Vogel-Fulcher form for the relaxation time by potulating a simple relationship between the free volume avable for structural relaxation and the corresponding viscoor relaxation time. By assuming a minimum possible frvolumev0 ~e.g., analogous to the free volume of a randoclose-packed system of hard spheres! one argues that therelaxation time will diverge;eO„1/(v f2v0)…, wherev f is theactual free volume available@12#. Linearizingv f(T) yieldsthe Vogel-Fulcher form. Of course, such an approach dnot take into account the nonequilibrium nature of the trsition. Nevertheless, the key idea of this phenomenologapproach, that relaxation on long length scales is frustraby the local structure~represented here by an average qutity, the free volume!, is appealing and appears in one foror another in most modern approaches.

Subtleties with the glass transition begin with definingprecise temperature at which it occurs. The glass transitemperatureTg can be operationally defined as the poupon cooling where the measured viscosity exceh'1013 P. Alternatively, one can heat a frozen glass aexamine the changing specific heat. In this case the g

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transition is defined by the onset temperature at which, inabsence of crystallization, the specific heat shows a pronent peak or jump@13#. One can also measure changing utrasonic acoustic properties as a glass is heated, in wcase the glass transition is indicated by a kink in a plotacoustic attenuation~or velocity! as a function of temperature @14,15#. Finally, one can measure the sample specvolume as a function of temperature@16#. This curve showsa kink at a temperature consistent with theTg predicted byacoustic attenuation or specific heat measurements. Thfore, provided the measurement time scale~MHz for theacoustic studies, effectively mHz for viscoelastic propertizero Hz for specific volume! is smaller than the experimentatime scale from the scanning rate@d(lnT)/dt#21, the glasstransition temperature appears to be frequency independ

The actual value ofTg depends on the time scale of thexperiment, i.e., heating or cooling rates, and on the samhistory ~annealing time at low temperature, for examp!consistent with the idea that the transition is a nonequirium phenomenon. An experimental study of the heating-rdependence ofTg and of the width of the specific-heat peahas been undertaken, with the aim of determining ifTg andthe heating rate are related by a Vogel-Fulcher law and iftransition width extrapolates to zero at a finite heating r@17#, with as-yet inconclusive results.

As one approachesTg from higher temperatures, glassforming materials show complex nonexponential decaysautocorrelation or relaxation functions. The time constat involved in these dynamic processes are strongly tempture dependent. The dependence oft(T) can often be ex-trapolated to show apparent divergences at nonzero temptures. Viscoelastic@18,19#, light scattering@20–22#, NMR@18,23,24#, ultrasonic@15#, specific-heat spectroscopy@25–27#, dielectric response@28–30#, and neutron-scattering@31–38# studies of highly viscous glass formers have shown coplex relaxational behavior. This is also evident in moleculdynamics simulations@39,40# of model systems that havshown that the relaxation process is quite complex. Eadielectric relaxation studies@30# of supercooled fluids indi-cated two distinct mechanisms, now known asa andb re-laxation. This was later confirmed in neutron-scatteri@34,36,38# studies and by molecular-dynamics simulatio@41,42#.

Loosely speaking,a relaxation is associated with longtime scales and long-range order at the glass transition, wthe fasterb relaxation involves shorter time scales andassociated with short-range order. Indeed, the transientcess,b relaxation, appears at temperatures aboveTg andpersists into the glass. Through study of ionic and polymeglasses above the glass transition this mechanism hasobserved to decay via power laws with temperatudependent exponents@20,34#. Depending on the time scaleconsidered, the corresponding relaxation times appeahave an Arrhenius temperature dependence (eA/kT, whereAis an activation energy andk is Boltzmann’s constant! or forsomewhat longer times an effective power-law form.temperatures are reduced~well aboveTg) the temperaturedependence of short-timeb relaxation follows an effectivepower-law dependence of the viscosityh;(T2T8)2m, withT8.Tg andm;2 @43#. The form is only an effective onesince as temperatures are reduced towardsT8, a relaxation,

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2134 55IAN S. GRAHAM, LUC PICHE, AND MARTIN GRANT

which involves Vogel-Fulcher temperature dependencdominatesb relaxation. The temperatureT8 marks a cross-over boundary between two different regions of viscoushavior @43#. This crossover occurs at a viscosity of arou102–103 P, a region readily accessible to other experimentechniques and to molecular-dynamics simulations. In fexperimental results have been obtained in neutrscattering studies@36,38#, while the two distincta and bmechanisms have also been observed in molecular-dynasimulations@41,42#. These experimental results onb relax-ation have been taken as confirmation of mode-couptheories, as discussed in two reviews@44,45#.

The long-time mechanism,a relaxation, is observed onlyat temperatures aboveTg . As the temperature is loweretowardsTg this feature dominates the late-time behavior ais characterized by nonexponential relaxation and divergeof the relaxation times near the glass transition temperatIt is the a mechanism that is most directly associated wlong-range order at the glass transition. At late times tdecay is often well described by a stretched-exponential~alsoknown as a Kohlrausch! decay

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with a,1, whereF(t) is a correlation function defined below. The stretch exponenta does not appear to be universand can vary from 0.2 to 0.8 depending on the material bestudied, the method of sample preparation, and in some cthe temperature. The appropriateness of this expressionbeen questioned@29# and other functions have been sugested, particularly in the context of dielectric response msurements@46#. However, the stretch form is widely usedue to its reliability in reproducing data from many differeexperiments and as a general scaling form. We reiterthough, that such a function is phenomenological.

As temperature is lowered and the glass transitionproached the relaxation timet or viscosity grows rapidly andin many cases extrapolates to a divergence at a temperT0,Tg . This divergence is often well fit by the expressio

t;eA/[k~T2T0!] , ~2!

known as the Vogel-Fulcher law whenT0.0. Some materi-als are best described withT050, i.e., Arrhenius relaxationwith activation energyA. Such materials typically formstrong covalently bonded networks, for example silicatMost glass-forming materials, such as polymers, metaglasses, and ionic systems do not form strong covalent boor bonded networks. These materials show non-Arrhenrelaxation near the glass transition, characterized by anparent divergence at a nonzero temperatureT0,Tg .

One question is whether there is a simple and univeform for a relaxation. Neutron spin-echo experiments sugest that nearTg the Kohlrausch decayce

2(t/t)a may satisfythis requirement with a rescaling of timest/t(T) @34,37#.These data also suggest that the prefactorc is temperatureindependent well aboveTg , but that it acquires temperaturdependence close to the transition~a nondiverging powerlaw!. These results are consistent with several molecudynamics simulations@41,42,47,48#, but not all@49#. Mean-while, the dielectric relaxation studies of Dixonet al. @25,29#

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suggest a more complicated universal form similar to tseen in multifractal scaling: There the Kohlrausch form donot appear to reproduce the high-frequency data. Othersults suggest that Dixon and co-workers’ scaling is not uversal and has to be modified to account for high-molecuweight polymers @28#. The discrepancy between thdielectric and neutron-scattering experiments has yet toresolved.

We can summarize as follows. Near the glass transitautocorrelation functions decay nonexponentially at ltimes bya relaxation, with the form often being well described by a stretched exponential. The corresponding reation times usually extrapolate to a divergence, viaVogel-Fulcher law, at a nonzero temperature below thatthe glass transition. Network glasses show the more straiforward Arrhenius form. There are also faster transieb-relaxation mechanisms that decay via power laws awhose relaxation times appear to diverge via Arrhenius laalthough when glass formers are studied at temperaturesabove the glass transition these mechanisms suggest etive power-law divergence of the viscosity at a temperathigher than the glass transition. A critical examinationmodels of the temperature dependence of glass-forminguids has been given recently by Stickelet al. @50#.

Theoretical investigations of glassy phenomena havelowed several approaches with mixed success. These incthermodynamic free-volume models@10#, blocking or diffu-sion models@51#, hierarchical models@52#, and hydrody-namic mode-coupling theories@44,45,53–57#. The oldest isthe free-volume approach discussed above. This idea of lstructure trapping the system in a metastable state hastested by several constrained dynamics models@3,51# and viahierarchical relaxation models@52#.

The former approach constructs an artificial dynamics tserves to slow relaxation. The results often yield nonexnential relaxation functions and in some cases a nArrhenius divergence of the relaxation time. However,general, the models are too artificial to be easily relatedreal systems. An exception is the facilitation kinetic Isinmodel. This model, introduced by Fredrickson and Ander@6,7#, attempted to surmount these problems by incorporadynamical frustrated relaxation into a simple microscomodel: the kinetic Ising model. Dynamical frustration is icorporated by allowing a spin to relax only when the neigboring spins are in an appropriate state, analogous to alling a flip only when a site is surrounded by sufficient locfree volume. Analytic study@6# of one version of this modesuggested power-law divergence of the relaxation time anonzero temperature and stretched-exponential decaysthe autocorrelation functions. Subsequent numerical wortwo dimensions verified stretched-exponential decay ofautocorrelation function, but found only weak evidencedivergence of the relaxation time at nonzero temperatuHierarchical models complement these constrained dynamapproaches. The model proposes a hierarchy of relaxaprocesses such that a state at a high level, correspondinglong relaxation time, can relax only if a certain set of staon lower hierarchy levels~with shorter relaxation time! havealso relaxed. This approach can yield Vogel-Fulcher laand stretch-exponential decays, given appropriate assutions about the relationship between the levels of the hie

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The above approaches are similar in that they consglassy phenomena to arise from frustration arising fromlocal structure. A conceptually different approach is hyddynamic mode-coupling theory@44,45,53–57#. This theoryhas met with some success, particularly in the moderaviscous region prior to the formation of a glass. The essenidea is that a glass arises from a hydrodynamic slowdown near the glass transition, where the slowing downgoverned by a nonlinear coupling between the density fltuations and the renormalized viscosity. The resulting mopredicts several different scaling regimes, two of which wmentioned above in the context of neutron-scattering resThese neutron results have been interpreted as a confirmof the predictions of mode-coupling theory. However, tagreement with experiment requires terms withinmemory kernel linking the density fluctuations and the renmalized viscosity that are difficult to justify theoretical@45#. As a result, any hydrodynamic mechanisms leadingglassy relaxation and the glass transition are obscured.

III. MODEL

The problem in simulating a glass is one of time, astime scale of atomic motion is many orders of magnitushorter than the time scale on which glassy phenomenaobserved. Thus, for example, molecular-dynamics calctions are limited to the moderately viscous regimeh'103 P,where precursor effects to glasses may be observed, bufar from the glassy regime ten orders of magnitude awMonte Carlo methods are similarly, though less restrictOur approach is slightly different: we postulate a dynamimechanism that gives rise to frustrated relaxation and glabehavior and incorporate it directly into the dynamics of tmodel. As a result, the effective time scale of our simulatiois on the scale of the frustration and not on the shorter tscale of atomic motion. We can therefore simulate mulater effective times and can test, by our results, whethenot our postulated mechanism is correct.

A reasonable candidate for a mechanism are local barto relaxation that depend on the local environment. Sbarriers will inhibit relaxation when the neighborhood is ttightly packed, in keeping with the frustration ideas dcussed above. In general, the height of these barriersdepend on the local environment. To simplify the modelassume only two possible barrier energies: zero or infinThis should retain the essence of the model while preserthe universal behavior.

We test this idea within a variant of the facilitation modof Fredrickson and Andersen, in both two and three dimsions. Following them, we consider a kinetic Ising modelthe absence of any site-site coupling (J50), but in the pres-ence of an external field. For convenience we set the extefield to unity ~or, equivalently, temperature is measuredunits of the strength of the external field!, defining a positivefield as being in the negative spin direction. The Hamiltonis

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wheres i561 is the spin at sitei andN is the number ofspins. There is no underlying thermodynamic phase tration; the system is always paramagnetic. In the absencany frustration rules the equilibrium and nonequilibriuproperties on the model are well known. All equilibrium ononequilibrium autocorrelation functions decay exponetially in time and the average spin at equilibriumm05^s&52tanh(1/kT). In addition, the spins are spatialluncorrelated, at all times, whether or not the system isequilibrium.

To add dynamical frustration, denoted facilitation, we aa rule such that a spin can attempt a flip~with standard Boltz-mann probability! only if the spin is surrounded by a sufficient number of up spins. Allowing the attempted flipequivalent to saying there is no barrier to the transitiowhile forbidding the flip implies an infinite barrier heighFor any lattice there can be several facilitation rules. A twsite facilitation rule is one where a spin flip is allowed onlytwo or more of the nearest-neighbor spins are in the spinstate. Thus the three-site rule requires a minimum of thup-spin neighbors, and so on. Such rules clearly serveslow the dynamics, and we can anticipate that the harderrule, the slower the evolution.

The kinetics of this model are easily implemented withMetropolis Monte Carlo dynamics by adding the facilitatiorule to the transition probabilities. They become

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whereEa52s i andEb5s i are the energies at sitei ~ini-tially with spin s i) before and after the attempted flip, respectively. Time is measured in units of Monte Carlo steper spin~MCS/spin!. Each Monte Carlo step correspondsN attempts to change the system’s state using this transprobablitity. The facilitation functionf ($s j%) depends onlyon the states of the spins$s j% neighboring the site of interesand is either 1 or 0 depending on whether or not theresufficient neighbors to satisfy the facilitation rule@59#. ~Con-sequently, distances are measured in units of the distabetween neighboring sites.! The effect of this rule is to slowthe dynamics in such a way that the equilibrium thermaverages are preserved. The glassy behavior we investhere is associated with this slow dynamics. This point hbeen discussed elsewhere@7#.

The facilitation model was incorporated within a multspin algorithm previously used to study equilibrium and noequilibrium properties of the kinetic Ising model@60#. Thisallows us to simulate 32 independent systems in paralleltwo dimensions we examined two-site facilitation, whilethree dimensions we studied both two-site and three-models. The advantage of our algorithm is its speedability to average over many independent systems. Thiimportant, as the phenomena we wish to study are attimes, where good statistics are necessary to extractfrom the background. Using our algorithm we are able

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Three different initial conditions were considered for onumerical studies:~i! we prepared a system initially with aspins up, quenched the system by applying a reversed fiand monitored the subsequent evolution;~ii ! we prepared thesystem in a magnetized equilibrium state and estimatedtime-dependent fluctuations around this state; and~iii ! weexamined the thermal fluctuating properties of glassy stwhich was prepared by quenching below the critical relation temperature~discussed below! and annealing the system for several 106 MCS/spin.

For the system quenched by the reversed fixed extefield 1/kT, we monitored the evolution of the average manetizationm(t)5^s& as the system evolved to equilibriumAt late times this magnetization reaches the equilibrivalue m`5 limt→`m(t). For sufficiently large systemm`[m0, so that the equilibrium average is equal to thpredicted by the equilibrium partition function. As noted prviously @7#, facilitation can trap the system within a restricted configuration space, provided the quench field isdeep for a given system size. In this casem`.m0. We usedthis criterion to detect finite-size effects in our simulatioand have accordingly increased system sizes to give coresults. Providedm`[m0, to within statistical error, we findthat other quantities we measure, in particular the relaxatimes, are independent of the size of the system. For dimsion d52, we used systems of sizeN5242–482, while ford53, we usedN583–163. Magnetizationsm(t) were aver-aged over anywhere from 640 to several thousand indedent quenches depending on the time required for the systo actually equilibrate. We then calculated the normalizrelaxation function

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In some cases we also calculated the spatial correlation ftion g(r ,t) during the quench and at equilibrium (t→`).This is defined by the average

g~r ,t !5 K ^s~r !s~0!&2m~ t !2

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where the inner averages^s(r )s(0)& andm(t) are takenover all appropriate spins for a given system and the oaverage is an ensemble average. In the absence of faction, g(r ,t)5d r50, since all sites are uncorrelated. Thshould also be the case in the presence of the facilitation,only at equilibrium, since facilitation should affect the noequilibrium and not the equilibrium properties of the systeWe examined equivalently the radially averaged power sptrum of the latticesS(q,t), which is the Fourier transform og(r ,t), suitably normalized in real space to giveS(0,t)[0.A purely uncorrelated system yields a flat power spectruwith the amplitude reflecting the noise strength~magnitudeof the fluctuations!.

For the systems prepared in equilibrium, we also msured the spin-spin autocorrelation function. In normalizform this is

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ways. We first considered slow uniform heating of the stem and measured during this process the effective speheat as a function of temperatureT. This is analogous to acalorimetric measurement of the glass transition. Secondconsidered the response of the system to a temperatureto higher temperature, monitoring in this case the avermagnetizationm(t) and the structure factorS(q,t). This is ofinterest with reference to a time-resolved x-ray studyamorphous metals@61#, which examined the early-time evolution of the structure factor.

IV. RESULTS

Typical results from quenching with the external reveing field are shown in Fig. 1 for different inverse tempertures 1/kT. These results are qualitatively independent ofdimensionality or of the choice of facilitation models: thonly apparent effect of making the facilitation more restrtive is to slow the dynamics. There are two clear regionsthese data: an early-time exponential, followed by late-tislower-than-exponential decay. The crossover from earlylate time is interesting: for a short time evolution slows s

FIG. 1. log-log plot of2 log10c(t), as a function of timetfollowing a quench from the all spin-up state, wherec(t) is therelaxation function. The results here are for a two-dimensional stem, using the two-site facilitation rule, at differing values1/kT as indicated. The results for other dimensions and modelsqualitatively the same. The arrowK marks the kink in the relax-ation function, as discussed in the text. The dashed line correspto ordinary exponential decaya51.

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nificantly, after which the decay begins again, but slowthan exponential. This behavior, indicated by the arrowK,can be explained as follows. At early times most of the spare up, so it is easy to flip them down. Here the facilitatirule plays almost no role, so decay is exponential. Eventuall the easy spins have been flipped and the remainingspins cannot flip until their environment has been arrangebring up spins into their neighborhood. This initial phaseup-spin diffusion is associated with very little change in taverage magnetization, since motion of spins requires sto be flipped up and then down: no net change in magnzation occurs. Therefore the decay inm(t) slows signifi-cantly. Eventually, however, clusters of up spins form aallow the decay to continue, albeit slower than exponentia

Evidence of this is seen in the dynamical evolution of tpair correlation function, shown in Fig. 2. At early and latimes the spins are uncorrelated, as expected, but at intediate times we see the buildup of short-range correlatiowhich dissipate as equilibrium is approached. The maximin this anomalous correlation takes place at the time wthe dynamical evolution is slowest. These correlations hapurely dynamical origin and disappear at late times asequilibrium state is achieved. Thus the facilitation rule donot introduce equilibrium correlations on any length scaleexpected.

We now turn to the behavior at equilibrium. In Fig. 3 wplot characteristic results for the equilibrium autocorrelatfunctions, shown here for a three-dimensional system wthe two-site facilitation rule. At low temperatures~highfields! the late-time behavior is well described by a streexponential, with the stretch slopea decreasing slightly withdecreasing temperature. This isa relaxation in the model.We fit the late-time data to the functione2(t/t)a to determinethe stretch powera and relaxation timet. To test the appro-priateness of this form we plotted ln(F) as a function ofln(t): there was no evidence for late-time power-law behior. We also fit the late-time data to the formce2(t/t)a, aswell as to the sum of ordinary exponentiace2t/t11(12c)e2t/t2. In the former case, the best-fit es

FIG. 2. Pair correlation functiong(r ) plotted for different timest following a quench from the all spin-up state. The buildupcorrelations at intermediate times is clearly evident. The maximamplitude of these correlations occurs at a time coincident withkink marked in Fig. 1.

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mates were most consistent withc.1, there being no systematic variation ofc with temperature. This is differenfrom hydrodynamic mode-coupling theory, which predictscusplike temperature dependence in this quantity. Compathe fits, we found that the stretch form represented the dbetter than the sum of exponentials. We can also estimaterelaxation timet from the integrated autocorrelation function. This gives an estimate entirely consistent with that otained from the fits, implying that the integral ofF(t) isdominated by a stretched exponential.

For times shorter than those shown in Fig. 3 the decaythe autocorrelation function are exponential. In the modthis time regime is what corresponds tob relaxation. Thedecay in this transient regime is due to fast motion of spthat are initially free to evolve, that is, are initially facilitateby a sufficient number of up neighbors. The slow noexponential late-time decay is due to spins that are initiablocked by the facilitation rule. Since the initial spin distrbution is random we can use a mean-field argument toculate the relaxation times of the initial transient exponenregime. At equilibrium the spin-up probability is given bf5e22/kT/(11e22/kT). Linear response around the equilibrium statef1de2t/tb yields 1/tb5pF( f )(11e22/kT), wherepF( f ), the probability that a given site is facilitated, is giveby the ratio

(i5 j

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. ~8!

HereN is the total number of neighbors andj is the mini-mum number of up neighbors needed for facilitation. Tagreement between the calculated relaxation timetb(T) and

eFIG. 3. log-log plot of2 log10F(t), as a function of timet for

temperatures, from left to right, of 1/kT 5 0.1, 0.2, 0.3, 0.4, 0.5,and 0.6, whereC(t) is the autocorrelation function. The resulhere are for a three-dimensional system, using the three-site fatation rule. Straight lines correspond to a stretched exponentialslope being the stretch powera. The dashed line correspondsordinary exponential decaya51.

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the fits to the early-time simulation data are essentially exand indicate Arrhenius behavior at low temperatures. Tothis analytically we need only consider the limit of smallf~i.e., low temperatures!, in which case the sums are domnated by their leading terms. The relaxation time is then

tb;e2 j /kT

~11e22/kT! j21'e2 j /kT, ~9!

which is an Arrhenius law. This is consistent with the intepretation of this short-time regime being analogous tofastb-relaxation mechanism seen in experiment.

Let us return to the nonexponential late-tima-relaxation regime and consider the fitted results forrelaxation timest and stretch powersa ~Fig. 4!. Fig. 4~a!shows the temperature dependence of these relaxation tfor the three models studied. A straight line implies Arrheius relaxation and divergence of the relaxation timeT50. All our results are concave upward, indicating divegence at a nonzero temperature. We fit these data topossible forms for this divergence, a power law andVogel-Fulcher expression, and have summarized the resin Table I. These Vogel-Fulcher fits are shown in Fig. 4~a!with solid lines and a dashed line shows the best ofpower-law fits; the other power-law fits are extremely poand are not shown~note thex2 values in Table I!. Consid-

FIG. 4. Plot of the~a! relaxation times and~b! stretch powers asa function of inverse temperature 1/kT, for the three cases studiedError bars~not shown! vary in size, from smaller than the displayesymbols to roughly twice the size of the symbols. A straight line~a! corresponds to Arrhenius relaxation. The concave upwcurves indicate divergence of the relaxation time at nonzero tperatures: the solid lines are the best fits using a Vogel-Fulcfunction with the parameters given in Table I, while the dashedis the best fit to a power law; the other fits are extremely poor.~b!indicates the possibility that the stretch power may be weaklypendent on temperature in the model.

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ering all three models, it is clear that the Vogel-Fulcher epression provides a much better fit to the data than doespower law, showing the intimate relation between stretchexponential time decay and the Vogel-Fulcher temperadependence within this model. The relaxation times obtaifrom the integral of the curves gives essentially identiresults. We have therefore shown that a microscopic, pudynamical model can give rise to this important aspectglassy phenomenology.

Figure 4~b! shows the temperature dependence ofstretch powera, wherea;0.5. Our results indicate a smasystematic increase ofa with temperature from about 0.35 t0.55 with temperature. Up to now the consensus has bthat the glassy regime is characterized by a sintemperature-independent stretch exponent. However,note that this weak temperature dependence ofa is extractedfrom fits, so there may be systematic error in our data ansis of which we are not aware. Further study would be nessary to conclusively demonstrate a temperature dedence ofa. We also note that this feature could be difficuto observe experimentally.

The similarity between the two regimes of the facilitatiomodel ~fast exponential and slow stretch exponential! andtheb- anda-relaxation mechanisms in true glasses is qusuggestive, although, in our simple model, we see no edence of the elaborate effective temperature dependeand power-law decays seen in experimentalb behavior. Thetransient exponentialb mechanism is due to fast relaxatiowithin a local, relatively free cage and the slower noexponentiala decay is due to slow diffusional relaxation othe trapped sites or particles. Thus our model for structuglasses reproduces both relaxation mechanisms presentthe glass transition and provides a physically reasonamechanism for both. We note that this physical argumentbeen previously discussed by Lewis@42#.

Finally, we comment on a possible universal scalingthese relaxation data. At first, the simple form indicatedFig. 4 would suggest that universal rescaling is possibHowever, the fact that the early-time data followe2t/tb(T)

precludes this possibility. Our data are consistent with tregimes, namely, the early-time exponential and the late-tstretched exponential. Naturally we cannot rule out otlate-time scaling forms. However, for the late-time regim

d-ere

-

TABLE I. Results from fitting ~a! Vogel-Fulcher(t5ceA/(kT2kT0)) and ~b! power-law (t5A(kT2kT0)

2g) expres-sions to the relaxation time data shown in Fig. 4. Thex2 values areweighted by the uncertainty in the data.

Model c A kT0 x2

~a!d52 ~two-site! 0.4360.02 4.260.1 0.5060.03 2.3d53 ~two-site! 0.1660.02 3.660.1 0.3060.01 2.1d53 ~three-site! 0.5160.02 5.260.2 0.6860.03 1.8

Model A g kT0 x2

~b!

d52 ~two-site! 5.360.5 1.360.1 1.1860.05 118d53 ~two-site! 1.860.1 3.260.1 0.5760.01 7.9d53 ~three-site! 506100 3.062.0 0.761.0 264

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we studied our data are consistent with a stretched expotial and are inconsistent with a power law.

These results give confidence in the use of this simmodel to investigate the glass transition. Hence we nextvestigated the thermal properties of model glass by quening a system below the divergence temperatureT0 ~indicatedin Table I!, relaxing it into a glassy state, and then slowheating it, measuring the average system energy perê& and fluctuationse2&2ê&2[12ê&2. The specific heatcan be obtained two ways: by differentiating the energyspin Cv5dê&/d(kT) or by the statistical expressioCv(st)5N(ê2&2ê&2)/(kT)2. The latter expression can be in

correct when the system is nonergodic, while the former wreflect the thermal properties as they would be measurean experiment. All these results can be contrasted withexact results for this model in the absence of facilitation.

To study the ergodicity, we used the two-dimensiontwo-site facilitation model. Results for a typical heating asubsequent cooling run are shown in Fig. 5, contrasted wthe exact Ising result~dotted line!. This system was initiallyquenched from the all spin-up state tokT50.454 55 andallowed to relax for 83106 MCS/spin. It was then heate~and subsequently cooled!, at a constant rate o131025kT/MCS. This initial temperature is below thT0'0.5 of this system~see Table I! so the relaxation time othe prepared glassy state is extremely long: even a83106 MCS/spin the system was still evolving very slowwith time. However, relaxation was so slow that no noticable change in properties would have occurred over the tscale of the heating-to-cooling run.

From the measurements of the specific heats, ergodion the time scale of our simulation measurements, bredown just belowkT52.0 for both heating and cooling runThe heating run shows a large peak in the specific h

FIG. 5. Specific heat as a function of temperaturekT for asystem heated fromT50.454 55 toT56 and subsequently cooleat a rateud(kT)/dtu5131025kT/MCS. The arrows indicate thedirection of the temperature change. The solid line gives the sttical specific heatCv

(st) defined in the text, while the open symbolinked by the dashed line give the experimentally relevant quanCv5dê&/d(kT). The dotted line is the exact result in the absenof facilitation. The large heating peak corresponding to a glass tsition is clearly evident, as is the breakdown of ergodicitykT<2 on cooling andkT<3 on heating.

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Cv , while the subsequent cooling run shows a much smaand broader peak centered at a slightly lower temperaturwe reduce the scanning rated(kT)/dt the large peak narrowsand moves to lower temperatures, while the broad coopeak is largely unchanged~results not shown!. This is alsoobserved with real glasses. The calorimetric behavior ofmodel thus possesses much of the characteristic phenenology of the thermally measured glass transition andposition of the maximum in the specific heat can be intpreted as the effective glass transition temperatureTg , whichwe will denote asTm below.

The quantityCv(st) ~solid line in Fig. 5! tells us about the

microscopic ramifications of the broken ergodicity assoated with this peak. Since this quantity diverges at low teperatures we see thatê2&2ê&2 does not go to zero aT→0. This means that fluctuations are quenched, on the tscale of the experiment, in keeping with dynamical argments for the origin of glassy behavior.

Also of interest are the results of scaling the specific-hdata for calorimetric experiments performed at differeheating rates. We analyzed such data using the standarding form, with results shown in Fig. 6. Provided the noneqlibrium peak is well displaced from the broad backgroupeak of the equilibrium model~i.e., it is at sufficiently hightemperatures! the data scale very well. This ceases to becase~dotted line! when a large portion of the specific heatdue to the equilibrium background~dotted line!. We notethat one can ensure that these two regions are well sepaby making the facilitation even more restrictive.

There has been interest in determining if, at asymptcally slow heating rates, the broad glass transition sharpup in the manner of true first- or second-order transitio@17#. This is usually done by measuring the glass transitpeak position and width, estimating their functional depedence over experimentally accessible heating and coo

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FIG. 6. Scaling plot for the glass transition. The data have bscaled according tok(T2Tm)/W, Cv /Cv

m , whereTm is the posi-tion of the midpoint of the peak,W is the full width at half maxi-mum, andCv

m is the peak height. The heating rates correspondto the different curves are ~a! 531023kT/MCS, ~b!131023kT/MCS, ~c! 531024kT/MCS, ~d! 231024kT/MCS, and~e! 131024kT/MCS. The dotted line~which does not scale! is thescaling result for systems heated at the very slow r531026kT/MCS.

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rates, and then extrapolating to slower rates. In our caseis equivalent to measuring the heating-rate dependence oscaling parameters, which we have plotted in Fig. 7.

Figure 7~a! shows the rate dependence of the peak ma

FIG. 7. Dependence of scaling parameters~a! Tm , ~b! peakwidth W, and ~c! specific-heat maximumCv

m on the heating rateResults are shown for two initial states annealed for differlengths of time atkT50.454 55: 83106 MCS/spin ~circles! and23106 MCS/spin~triangles!. The dashed lines in~a! are best fits toan Arrhenius law, while in~b! they correspond to power-law fits, adiscussed in the text.

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i-

mum Tm : the Arrhenius form is clearly evident. SimilaArrhenius behavior for the glass transition temperature isten seen in typical calorimetric experiments@25,62# and im-plies that the effective activation energy arising from tdynamical frustration is independent of the heating rate.check that this is a function only of the frustration mechnism and not of the annealed state we repeated the numesimulations, starting with an initial state annealed for a mushorter time atkT50.454 55 (23106 MCS/spin as opposedto 83106 MCS/spin!. The results, also shown in Fig. 7~a!,show that the effective activation energy is largely indepdent of the initial state of the system.

Also of interest is the rate dependence of the peak widplotted in Fig. 7~b!. We have attempted to fit these data vpower series in heating rate or by a power law with anditive constantW5A1Bta. The power law provides by fathe better fit and yields a power-law growth exponent forpeak width of approximately 0.6. More importantly, thpower law extrapolates to a finite-width transition at sloheating rates. Of course this extrapolation only makes sefor time scales shorter than the annealing time of the ppared low-temperature system. This also applies to realperiments.

In Fig. 7~c! we plot the specific-heat maximum as a funtion of heating rate. In general, the maximum increases wdecreasing heating rate, until a critical rate is reached whupon the maximum begins to decrease. The value ofmaximum, as well as the crossover point between theseregimes, depends on the annealing time of the starting sA longer annealing time yields larger specific-heat maxiand at the same time lowers the heating rate at whichcrossover from increasing maxima to decreasing maximseen. We also examined the annealing time dependencexamining, at a fixed heating rate of 531024kT/MCS, theeffective glass transition peak for system annealed for timbetween 43105 and 83106 MCS/spin. We observe that thmaximum in the specific heatCv

m and the temperature of thipeakTm both increase with increasing annealing time, whthe peak widthW decreases~data not shown!. In addition,the integral over the peak estimated by the productCv

mWincreases with aging. This is quite consistent with expemental observations of typical glasses. The numerical dare, however, too noisy for us to be able to estimatefunctional form of these dependences.

Finaly, we examine the response of the glass state toupward temperature jump. For this study we took configutions that had been annealed for 83106 MCS/spin atkT50.454 55, instantaneously jumped the temperaturehigher value, and monitored the subsequent time evolutiothe magnetization and the ensemble-averaged structure fS(q,t). Jumps were performed to temperatures betwkT51.0 andkT56.0. Characteristic results are for magnezation and extreme-q components of the structure factor ashown in Fig. 8 for a jump tokT54.0, while Fig. 9 shows athree-dimensional topographic map of the evolution ofstructure factor with time.

The results can be summarized as follows. As the glstate evolves towards equilibrium the structure factor itially develops a peak atq50, which subsequently decayMeanwhile, the structure factor at largeq evolves monotoni-

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cally towards the equilibrium value. Theq50 peak is clearlydue to fluctuations in the magnetization: in fact, visualizatof the spin configurations during the jump show the formtion of domains of the fluctuating, high-temperature phaembedded within the still-frozen glass. The broad peak inences the structure factor out to approximatelyq50.35, i.e.,on the order of three inverse lattice spacings. Since eachin our model reflects an averaged domain in a real systthis implies quite large minimum domain sizes at the glapeak.

The peak atq50 is strongly reminiscent of structure factor evolution in spinodal decomposition when the orderrameter is not conserved@63#. This is consistent with ourmodel, which does not conserve magnetization. Howethe dynamics of our model will be quite different from ordnary spinodal decomposition, since in the latter case thenamics are driven by local free-energy gradients, whileour case the dynamics arise solely from the frustration.

We attempted to quantify this study by measuring theqdependence of the relaxation timest extracted from the

FIG. 8. Time dependence of the smallest- and largest-q compo-nents of~a! the structure factorS(q,t) and ~b! the average magnetization for a temperature jump fromkT50.454 55 tokT54.0. Thedashed line shows the result for largeq, namely,q50.927, in unitsof reciprocal lattice spacing. The solid line is for the smallestq,namely,q50.046 247.

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S(q,t) curves. Unfortunately, our data are too noisy andnot extend to small enoughq for this to be feasible. We findthat the rise inS(q,t) at early times is best described bypower law, albeit with a power-law exponent that variwidely with q, but nonsystematically. Also, the drop inSfollowing the peak is described neither by a power law nan exponential: the best description comes from a WigfunctionS;tue2t/t. Meanwhile, at highq the structure fac-tor grows almost linearly with time to the equilibrium valu

These results, however, are in qualitative agreement wthe time-resolved x-ray diffraction study of Brauer@61#. Inthat temperature-jump study of glassy Co2B and Fe76B24Brauer found, prior to crystallization, a rapid rise in thstructure factor at lowq, but no equivalent rise at highq,similar to our results discussed above. In Brauer’s case, hever, this initial growth is exponential, allowing one to etract relaxation times as a function ofq. Brauer foundt}q2, consistent with the rise being due to density fluctutions. In addition, Brauer found the rapid rise inS(q,t) onlyfor q,1.231022 Å 21, corresponding to a length scale o;80 Å. This is consistent with our findings that the smallelength scale for the density fluctuations should be ratlarge.

We performed temperature jump experiments to variotemperatures betweenkT51.0 and 6.0. As with the glastransition data we attempted to scale these temperature jresults onto a universal curve. This proved unsuccessfulboth the evolution of the structure factor evolution and tevolution of the average magnetization.

V. DISCUSSION

In this paper we presented and studied a facilitated kinIsing model for structural glasses. This model, a simplifiction of the model of Fredrickson and Andersen@6,7#, consid-ers a system of noninteracting Ising spins in a negativeternal field. Glasslike frustration, or facilitation, is introduceby allowing the spin at a given site to evolve via Metropodynamicsonly if it is surrounded by a sufficient number oup spins. If there are insufficient up neighbors the sitefrozen and does not evolve. As the field increases~corre-sponding to decreasing temperature! the equilibrium statecontains fewer and fewer up spins and more spins are froby the facilitation rule. This is analogous to saying that co

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FIG. 9. Three-dimensional topographic ploof the structure factorS(q,t) following a tem-perature jump fromkT50.454 55 tokT54.0.The contour plot in theq-t plane shows the structure factor peak atq50. The structure factor alate time is flat, indicating systems at equilibriumS(q,0), however, is slightly depressed at lowq,reflecting the residual structure of the nonequilirium glass.

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tightly packed regions of solid inhibit structural relaxationtheir neighbors. At sufficiently low temperatures the facilittion requirement should frustrate relaxation completely,which case the relaxation time should diverge and the sysshould become a glass.

The experimentally observed properties of glass-formmaterials near but above the glass transition can be sumrized as follows. There is an early-time power-law relaxatmechanism (b relaxation!, which persists into the glass anhas a temperature-dependent relaxation time that followArrhenius law, for some regime of temperature and timThis relaxation mechanism is believed to be associatedfast local atomic rearrangements. At late times there isecond nonexponential relaxation mechanism (a relaxation!that is often well described by a stretched exponential. Tcorresponding relaxation times are observed to diverge wtemperature following a Vogel-Fulcher laweA/k(T2T0), witha nonzeroT0,Tg . This is a diffusional relaxation mechanism and it is the cessation of this diffusion that allowsglass to form.

The results from the facilitation model are entirely cosistent with these observations. We first note that theretwo well-defined relaxation regimes. At short times the eqlibrium autocorrelation function decays exponentially, wan associated relaxation time that follows an Arrhenius laThis relaxation is associated with local~fast! spin relaxation,but not spin diffusion. At later times the relaxation takesa stretched-exponential form, with a stretch exponenta thatdepends no more than slightly on temperature and a prefathat is temperature independent. In this case we measurerelaxation times and tested their dependence on temperaIn all cases studied we found their divergence best descrvia a Vogel-Fulcher law. This slow relaxation is due to tslow diffusion of up spins, which then facilitate the relaation of otherwise frozen sites.

Given the success of the model reproducing these featof glasses, we turned to other typical experiments. Wemodeled calorimetric experiments of glasses by heatinfrozen model glass at a constant rate and measuringevolved nonequilibrium heat. In an experiment the glatransition is marked by a prominent peak or jump in tnonequilibrium specific heat under these conditions. Ourmerical results are consistent with these observations, shing a prominent peak in effective specific heat that canattributed to a glass transition. This peak is entirely duenonequilibrium behavior. As the heating rate is lowered tpeak narrows and moves to lower temperatures. This isconsistent with experiments on structural glasses. In ation, our model indicates a universal scaling of the noneqlibrium specific-heat curves for different heating rates.our knowledge, such scaling has not been attemptedexperimental data.

Finally, we turned to temperature jump experiments ameasured the evolution of the structure factor of a frozglass following a jump to temperatures aboveTg . At largeq the results show a monotonic increase of the structuretor towards the equilibrium value. The low-q data, on theother hand, show the formation of a large peak atq50,which then disappears as equilibrium is approached. As mtioned above, these results are qualitatively consistent wthe time-resolved x-ray measurements of Brauer@61#. How-

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ever, our data are too poor to allow qualitative measuremof relaxation times.

The agreement between our model and experiment iscomplete. For example, our model does not reproducecomplicated transient time and temperature dependencthe b-relaxation regime. Furthermore, within tha-relaxation regime, our model indicates a temperatuindependent prefactorc5O(1), different from the predic-tions of mode-coupling theory and from some experimenFinally, the effective thermodynamics upon heating a moglass are only qualitatively similar to data on real glasse

However, despite these drawbacks, the agreementtween the numerical results of our model and the knophenomenology of glasses is quite remarkable, especconsidering the extreme simplicity of our noninteractingnetic Ising model. We therefore believe that the model ctains the essential mechanism of glassy phenomena, namrelaxation frustrated by local structure. In such models relation on long length scales~and hence long time scales! canonly occur by the diffusion of up spins through the systeMeanwhile, the diffusion of these spins is frustrated by tpaucity of up-spin neighbors. This hierarchical proceserves to slow the diffusion and, for sufficiently few uspins, effectively stop it.

There are several remaining issues. First, we note thatmodel considers the facilitation energy barrier to be eitzero ~the site is facilitated! or infinity ~the site cannotevolve!. Thus the energy surface is filled with infinitely hig~or zero-height! barriers separating different spin configurtions. However, one could reasonably assume differentergy barriers depending on the degree of facilitation. Twould change significantly the shape of the energy surfaadding barriers of various heights. We note that the approof Fredrickson and Andersen@6,7# includes such a modification to the energy surface. In their model the insufficienfacilitated sites never evolve~infinite barrier!, while thosethat can evolve have their flipping probabilities weightedthe number of spin-up neighbors@59#. In this case the simu-lation results did not show the rich glasslike phenomenolowe observe. In light of the success of our simpler modereinvestigation of this more general approach would seembe in order.

We have also ignored the role of the conservation lawour treatment the magnetization was nonconserved. Thconsistent with structural glasses, but may not be consiswith network glasses, where the number of network bondconserved. The conservation of this quantity, in the contof our model, may explain why network glasses make sustrong glass-forming materials and also why network glasshow Arrhenius, not Vogel-Fulcher, divergences of thslow relaxation times. Simulation studies of variants of tfacilitation model may answer some of these questions.

Finally, we point out the need for an analytic study. Sinthe facilitation model appears to encompass the relevmechanism for glasses, studies of this model should slight on the analytic properties of glassy relaxation. In adtion, we suspect that the hydrodynamic approach, takenthe appropriate limit, will map onto a facilitation model. Thmight prove a useful way of explaining the origins of certaterms in the mode-coupling memory kernel.

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ACKNOWLEDGMENTS

We wish to thank Dr. K. Elder, Dr. R. Bru¨ning, and Dr. L.Lewis for useful discussions and Professor D. Staufferuseful comments on a preliminary version of this paper. T

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work was supported by the Natural Sciences and Engining Research Council of Canada, les Fonds pour la Fortion de Chercheurs et l’Aide a` la Recherche de la Provincdu Quebec.

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@1# F. Spaepen and D. Turnbull, Annu. Rev. Phys. Chem.35, 241~1984!.

@2# G.B. McKenna, inComprehensive Polymer Science VolPolymer Properties, edited by C. Booth and C. Price~Perga-mon, Oxford, 1989!.

@3# J. Jackle, Philos. Mag. B56, 113 ~1987!.@4# J. Adler, Physica A171, 453 ~1991! reviews the related topic

of bootstrap percolation.@5# H. Vogel, Phys. Z.22, 645 ~1921!; G.S. Fulcher, J. Am. Ce-

ram. Soc.8, 339 ~1925!.@6# G.H. Fredrickson and H.C. Andersen, Phys. Rev. Lett.53,

1244 ~1984!; J. Chem. Phys.83, 5822~1985!.@7# G.H. Fredrickson and S.A. Brawer, J. Chem. Phys.84, 3351

~1985!.@8# H. Hakanishi and H. Takano, Phys. Lett. A115, 117 ~1986!.@9# A preliminary account of this work has been published in

short paper: I. Graham, L. Piche´, and M. Grant, J. Phys. Condens. Matter5, L349 ~1993!.

@10# D. Turnbull and M.H. Cohen, J. Chem. Phys.34, 120 ~1961!.@11# M.H. Cohen and G.S. Grest, Phys. Rev. B20, 1077 ~1979!;

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@59# In the Fredrickson-Andersen model the facilitation functidepends on the actual number of neighboring up spinsmj . For

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ping probability, when two or more neighbors have up spinsweighted by the number of up-spin neighbors. By choosf5$0,1% we preserve the essence of the model: that spins winsufficient up-spin neighbors cannot evolve and have mesimplified the microscopic dynamics for those that do evolThe advantage is that the extrapolation to a zero-siteyields the kinetic Ising model, which is not the case for toriginal Fredrickson-Andersen model.

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Model for dynamics of structural glasses

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