A Topographic View of Supercooled Liquids and Glass Formation Author(S): Frank H

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A Topographic View of Supercooled Liquids and Glass Formation Author(s): Frank H. Stillinger Source: Science, New Series, Vol. 267, No. 5206 (Mar. 31, 1995), pp. 1935-1939 Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/2886441 Accessed: 31/03/2010 22:44

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Hall and P. G. Wolynes, J. Chem. Phys. 86, 2943 this event for very many such molecules, and this Soltwisch, D. Quitmann, Phys. Rev. Lett. 69, 1540 (1987). has the character of a high positive entropy (1992). 110. H. Frauenfelder, S. G. Sligar, P. C. Wolynes, Sci- change, small (net) negative volume change, 139. L. Borjesson, A. K. Hassan, J. Swenson, L. M. ence 254, 1598 (1991). chemical conversion process. Torell, Phys. Rev. Lett. 70, 4027 (1993). 111. W. Doster, A. Bachleitner, R. Dunau, M. Hiebi, E. 121. V. N. Morozov and T. Ya Morozova, J. Biomol. 140. A. K. Hassan, L. B13rjesson,L. M. Torell, J. Non- LOscher,Biophys. J. 50, 213 (1986). Struct. Dyn. 11, 459 (1993). Cryst. Solids 172-174, 154 (1994). 112. V. I. Goldanskii, Yu. F. Krupyanskii,V. N. Fleurov, 122. R. C. Hoseney, K. Zeleznak, C. S. Lai, Cereal 141. J. Colmenero, A. Arbe, A. Alegria, Phys. Rev. Lett. Dok. Akad. Nauk SSSR 272, 978 (1983). Chem. 63, 285 (1986). 71, 2603 (1993). 113. F. Parak, J. Heidemeier, G. U. Nienhaus, Hyperfine 123. A. Zipp and W. Kauzmann, Biochemistry 12, 4217 142. K.-L. Ngai, J. Chem. Phys. 98, 7588 (1993). Interactions 40, 147 (1988). (1973). 143. , in Diffusion in Amorphous Solids, H. Jain 114. W. Doster, S. Cusack, W. Petry, Nature 337, 754 124. A. Petry et al., Phys. B. Condens. Mater. 83, 175 and D. Gupta, Eds. (The Minerals,Metals, and Ma- (1989). (1991). terials Society, Warrendale, PA, 1994), p. 17. 115. B. F. Rasmussen, A. M. Stock, D. Ringe, G. A. 125. B. Frick and D. Richter, Phys. Rev. B 47, 14795 144. M. H. Cohen and D. Turnbull,J. Chem. Phys. 34, Petsko, ibid. 357, 423 (1992). (1993). 120 (1960). 116. K. Kuczera, J. Smith, M. Karplus,Proc. Natl. Acad. 126. B. Frick, D. Richter, W. Petry, U. Buchenau, Z 144. P. Harrowell,private communication. Sci. U.S.A. 87,1601 (1990). Phys. B70, 73 (1988). 1 17a.1. V. Sochava, G. I. Tseretoli, 0. I. Smirnova, Biofiz- 127. A. Chahid, A. Alegria, J. Colmenero, Macromole- 145. T. Atake and C. A. Angell, J. Phys. Chem. 83, 3218 ika, 36 (1991). cules 27, 3282 (1994). (1979). 1 1 7b.l. V. Sochava and 0. I. Smirnova, Food Hydrocol- 128. B. Frickand D. Richter, Science 267, 1939 (1995). 146. S. Cusack and W. Doster, Biophys. J. 58, 243 loids 6, 513 (1993). 129. U. Buchenau and R. Zorn, Europhys. Lett. 18, 523 (1990). 118. G. Sartor, A. Hallbrucker, K. Hofer, E. Mayer, J. (1992). 147. C. Herbst et al., J. Non-Cryst. Solids 172-174, 265 Phys. Chem. 96, 5133 (1992); G. Sartor, E. Mayer, 130. J. Shao and C. A. Angell, unpublished results. (1994). G. P. Johari, Biophys. J. 66, 249 (1994). 131. L. V. Woodcock, Chem. Phys. Lett. 10, 257 (1971). 148. L. Haar, J. Gallagher, G. Kell,G.S. National Bureau 119. J. L. Green, J. Fan, C. A. Angell, J. Phys. Chem. 98, 132. A. Rahman, R. H. Fowler, A. H. Narten, J. Chem. of Standards-National Research Council Steam 13780 (1994). Phys. 57, 3010 (1972). Tables (McGraw-Hill,New York, 1985). 120. K. E. Prehoda and J. L. Markley, in High Pressure 133. J. Kieffer and C. A. Angell, J. Non-Cryst. Solids 149. P. W. Anderson, in III-Condensed Matter, R. Bal- Effects in Molecular Biophysics and Enzymology, 106, 336 (1988), and other references cited therein. lian, R. Maynard,G. Toulouse, Eds. (North-Holland,; J. L. Markley, C. A. Royer, D. Northrop, Eds. (Ox- 134. C. Boussard, G. Fontaneau, J. Lucas, J. Non- Amsterdam, 1979), pp. 161-261. ford Univ. Press, New York, in press), any refer- Cryst. Solids, in press. 150. A. L. Greer, Science 267, 1947 (1995). ences cited therein, and J. L. Markley, private 135. R. J. Roe, J. Chem. Phys. 100, 1612 (1994). 151. I would like to acknowledge the support of the communication. Note the important reassess- 136. A. J. Martinand W. Brenig, Phys. Status Solidi 64, NSF-DMR under Solid State Chemistry grant ment of packing efficiencies of folded and unfold- 163 (1974). DMR9108028-002, and the help of many col- ed states which is made in this study. While the 137. E. Duval et al., Phys. Rev. Lett. 56, 2052 (1986); A. leagues through stimulating discussions of this unfolding transition may, like the melting of a small Boukeneteretal., ibid. 57, 2391 (1986); E. Duvalet subject area. In particular, I am grateful to U. crystal, be a transition between two global minima al., J. Phys. Condens. Mater. 2,10227 (1990). Buchenau, H. Frauenfelder, T. Grande, W. Kauz- for the individual molecules, the phenomenon ob- 138. V. N. Novikov and A. P. Sokolov, Solid State Com- mann, W. Kob, P. Madden, J. Markley,P. McMillan, served in a protein solution is the consequence of mun. 77, 243 (1991); A. P. Sokolov, A. Kislink,M. P. Poole, H. Sillescu, F. Sciortino, and G. Wolf.

A Topographic View of Interaction Potentials Condensed phases, whether liquid, glassy, Supercooled Liquids and or crystalline,owe their existence and mea- surable propertiesto the interactionsbe- tween the constituentparticles: atoms, ions, Glass Formation or molecules.These interactionsare com- prised in a potential energy function FrankH. Stillinger ..(r ... . rN) that depends on the spatial location ri for each of those particles.The Various static and dynamic phenomena displayed by glass-forming liquids, particularly potential energyincludes (as circumstances those near the so-called "fragile" limit, emerge as manifestations of the multidimensional require) contributions from electrostatic complex topography of the collective potential energy function. These include non- multipoles and polarizationeffects, cova- Arrhenius viscosity and relaxation times, bifurcation between the a- and 1-relaxation lency and hydrogen bonding, short-range processes, and a breakdown of the Stokes-Einstein relation for self-diffusion. This mul- electron-cloud-overlaprepulsions and long- tidimensional viewpoint also produces an extension of the venerable Lindemann melting er range dispersion attractions, and in- criterion and provides a critical evaluation of the popular "ideal glass state" concept. tramolecularforce fields. Obviously, the chemicalcharacteristics of any substanceof interest would substantiallyinfluence the details of (D. Time evolution of the multi- Methods for preparingamorphous solids tunately, focused and complimentaryef- particlesystem is controlledby the interac- include a wide rangeof techniques.One of forts in experiment,numerical simulation, tions, and for most applicationsof concern the most prominent,both historicallyand and analytical theory currentlyare filling here the classicalNewtonian equationsof in currentpractice, involves cooling a vis- the gaps. motion (incorporatingforces specified by cous liquidbelow its thermodynamicfreez- The presentarticle sets forth a descrip- (D) providean adequatedescription of the ing point, through a metastable super- tive viewpointthat is particularlyuseful for particledynamics. cooled regime, and finally below a "glass discussingliquids and the glassesthat can In orderto understandbasic phenome- transition"temperature Tg. A qualitative be formedfrom them, althoughin principle na related to supercooling and. glass for- understanding,at the molecularlevel, has it appliesto all condensedphases. Concep- mation, it is useful to adopt a "topograph- long been availablefor materialsproduced tual precursorsto this viewpoint can be ic" view of (D. By analogy to topographic by this latter preparativesequence; how- found throughoutthe scientific literature maps of the Earth'sfeatures, we can imag- ever, many key aspectsof a detailed quan- (1), but most notably in the workof Gold- ine a multidimensionaltopographic map titative description are still missing. For- stein (2). The objective here is to classify showing the "elevation" (D at any "loca- and unify at least some of the many static tion" R (r1 ... rN) in the configuration The author is at AT&TBell Laboratories, MurrayHill, NJ and kinetic phenomenaassociated with the space of the N particlesystem. This simple 07974, USA. glass transition. change in perspectivefrom conventional

SCIENCE * VOL. 267 * 31 MARCH 1995 1935 three-dimensional space to a spaceof much predict a reliably from known molecular higher dimension[3N for structurelesspar- structuresand interactions. Transitionstates ticles, and even more for particlesthat are Basin (saddlepoints) asymmetricor nonrigidor both (3)] intrin- Melting and Freezing Criteria sically creates no new information,but it facilitatesthe descriptionand understanding The topographicview of the (D-scapeadvo- 0) of collective phenomena that operate in cated above leads to a clean separationbe- e~~~~~~~~~~ condensedphases, particularly in liquidsand tween the inherentstructural aspects of the Amorphous glasses. substanceunder consideration(that is, the inherent 0 An obviousset of topographicquestions classificationof potentialenergy local mini- Q. structures to askconcems the extremaof the 1Dsurface, ma), and the "vibrational"aspects that con- such as maxima("mountain tops"), minima cem motions within and among the basins Crystal ("valley bottoms"), and saddle points definedby those inherentstructures. Such a permutations ("mountainpasses"). The minima corre- separationleads naturallyto a fresh exami- spond to mechanicallystable arrangements nationof an old but veryuseful idea, namely, Particle coordinates of the N particlesin space, with vanishing the Lindemannmelting criterion for crystal- Fig. 1. Schematic diagram of the potential energy forceand torqueon everyparticle; any small line solids,first formulated in 1910 (6). hypersurface in the multidimensional configura- displacementfrom such an arrangement The Lindemanncriterion concerns the tion space for a many-particle system. gives rise to restoringforces to the undis- dimensionless ratio, f(T), of the root- placedarrangement. The lowest lying mini- mean-square (rms) vibrational displace- ma arethose whoseneighborhoods would be ment of particlesfrom their nominal lattice selected for occupationby the system if it positions,to the nearest-neighborspacing a. It assertsthat when rise causes Supercooled E i were cooled to absolutezero slowly enough temperature liquid liquid to maintainthermal equilibrium; for a pure f(T) to reach a characteristicinstability substance,this wouldcorrespond to a virtu- value, melting occurs. X-ray and neutron . // allyperfect crystal. Higher lying minima cor- diffraction experiments provide measure- respondto amorphousparticle packings and ments of i(T) (throughthe Debye-Waller aresampled by the stableliquid phase above factor)for a wide varietyof real substances, E the meltingtemperature Tm (4) and computersimulations can be used to Figure1 shows a highly schematic illus- calculate f(T) for model systems. These tration of the multidimensional '4)- results show that f(Tm) varies a bit with ~Crystal scape." Such a simplified representation crystalstructure: It is approximately0.13 for can be misleading, but it serves to stress face-centered-cubiccrystals and 0.18 for several key points. First, the minima have body-centered-cubiccrystals (7). In any 0 Tm a substantial variation in depth and are event, it is substantiallyconstant across sub- Temperature arrangedin a complex pattern throughout stancesin a given crystalclass and provides of the Fig. 2. Root-mean-square particle displacement configuration space. Second, each mini- a good account pressuredependence divided by mean neighbor separation, versus tem- mum is enclosed in its own "basin of of Tmfor a given substance. perature, for crystal and liquid phases. The value of attraction,"most simplydefined as the set Vibrational motions contributing to this ratio for the crystal at the melting point, f(Tm), is of all configurationsin its "valley,"that is, f(Tm) have a significantanharmonic char- specified by the Lindemann melting criterion. all locations that strict downhill motion acter.But asidefrom a verysmall concentra- would connect to that minimum. Third, tion of thermallycreated point defectsin the contiguous basins share a boundarycon- crystal,these vibrationsare confinedto the devisedto measuref(T) for liquids.Never- taining at least one saddle point, or tran- basinssurrounding the N! absoluteminima. theless, computersimulations for modelsof sition state. Fourth, equivalent minima At any temperatureT, then, the Lindemann real substancescan be designed to supply can be attained by permutationsof iden- ratiofor the crystalcan be expressed: the needed information.Numerically, these tical particles.This last point implies,for a simulationsare requiredto generatea rep- f(T) = <(AR)2>112/(Na) (2) pure substance, that every minimum be- resentativecollection of system configura- longs to a group of N! equivalent minima where AR is the intrabasinvibrational dis- tions for the temperatureof interestand to distributedregularly throughout the mul- placement from the absolute minimumin evaluatethe rmsparticle displacements that tidimensionalconfiguration space. the multidimensionalconfiguration space, returneach configurationto its correspond- An important issue concerning the and the bracketsdenote a thermalaverage ing inherent structure.Although only a (D-scapetopography that remains largely confined to that basin at temperatureT. small number of simulationsof this kind unresolvedconcerns the numberof minima Although it is not obvious in the usual have thus farbeen carriedout (8), the main fl(N), and in particularhow fast it rises way of presentingthe Lindemannmelting featuresof this extension are clear, and are with N. Rather general arguments (bol- criterion,Eq. 2 has a straightforwardexten- summarizedqualitatively in Fig. 2. The steredby exact calculationsfor some special sion to the liquidphase. One simplyrecog- crystaland liquid f(T) curves are distinct; theoretical models) yield a simple generic nizes that the thermalaverage and the dis- both monotonicallyincrease. with T, with form for the large-Nlimit in a single-com- placementsrefer to inherent structuresof the curve for the liquid located well above ponent system (4, 5): the amorphous packing and their associated that for the crystal.At the melting-freezing 4)-scape basins that predominate after melt- point, F'iq is approximately three times that - N! Q(N) exp(aN) (1) ing. The mean nearest-neighbor distance a for the crystal; equivalently, the rms parti- where a > 0 depends significantly on the for the liquid phase (obtained from the mea- cle displacement is approximately one-half chemical nature of the substance consid- sured radial distribution function) typically that of the nearest-neighbor spacing. ered. The permutational factor N! has al- is close to that of the unmelted crystal. In its conventional form, the Lindemann ready been explained; the challenge is to No laboratory experiment has yet been criterion advances an asymmetric, one-

1936 SCIENCE * VOL. 267 * 31 MARCH 1995 FRONTIERS IN MATERIALS SCIENCE: ARTICLES

of the systemat temperatureT corresponds Fulcher(VTF) form (10): to preferentialoccupation of basins with -To exp [A/(T - To)] (4) depth +*(T), which is the + value that T,(T) maximizesthe simple combination(5, 9): where To is in the picosecondrange and A and To are positive constants. u(0) - (kBT)-1[+ fj(+,T)] (3) So long as all TVare substantiallyshorter where kBis Boltzmann'sconstant. When T than the time available for experimental 0 is near Tm,this combinationhas two local measurement,the supercooled liquid re- maxima with respect to +; the first-order mainsin a state of quasi-equilibrium,and in to a discon- inhabitsand moves amongbasins 0. melting transitioncorresponds particular tinuous change in +8 as the role of the whose depths are closely clusteredaround T the mean relax- 0 fT absolutemaximum switches from one to the (1q(T). But as declines, other at TM. ation times representedby the VTF form Supercoolingthe liquidphase below Tm increasestrongly and all crossthe time scale kinetically avoids switching back to deep of experimentalmeasurement at essentially Tg"1To-1 crystallinebasins, but ratherthe liquid re- a common glass transitiontemperature Tg Inverse temperature mainsin those that correspondto the other > To, Furtherreduction in T fails to lower inhabited basins below Fig. 3. Temperature dependence of peak relax- local maximumof Eq. 3, given by fNiq(T), the depth of the ation (or absorption) frequencies for glass-forming that continuesto referto higherlying amor- (jIiq(Tg).The supercooledliquid then has liquids. phous inherent structures.So long as the fallen out of quasi-equilibrium.The ratio systemconfiguration point R(T) can move Tg/Tmfor most good glass-formingliquids more or less freely among the higher lying falls in the rangefrom 0.60 to 0.75. phase view of the first-order crystal-liquid amorphousinherent structures and attain a Careful examination of the relaxation transition. This model contrasts markedly representative sampling while avoiding functions gv(t) above Tg reveals the pres- with the thermodynamic description that crystalnucleation, the systemremains in a ence of distinct processes.At very short calls for two-phase equality of pressure and reproduciblequasi-equilibrium state of liq- times (comparableto vibrationalperiods), chemical potential at the transition. How- uid supercooling(9). intrabasin relaxation predominates.This ever, the extension just described effectively The individualtransitions that carrythe domainis followedby a much moreextend- restores two-phase symmetry. It supplements systembetween contiguous (boundary-shar- ed time regime during which interbasin the melting criterion with an exactly anal- ing) basins apparentlyalmost always in- structuralrelaxation processes occur, and in ogous Lindemann-like freezing criterion for volve localizedparticle rearrangements. In the long-timelimit of this regimethe relax- the liquid, specifically, that thermodynamic otherwords, only order0(1) out of N of the ation~inevitably seems to display a Kohl- instability with respect to freezing occurs particlesundergo substantial location shifts. rausch-Williams-Watts(KWW) "stretched when cooling causes fliq(T) to decline to This is truewhether the basinsare those for exponential"decay (11): the cited transition value. substantiallycrystalline inherent structures g,(t) - Fr exp [-(t/t)0] (5) Although it is relatively difficult to su- (and the transitioninvolves creationor de- perheat crystals above their Tm, supercool- structionof point defects),or whetherthey in which 0 < 0 ' 1 and FVis a constant; ing of the melt is commonplace, particularly referto a pairof amorphouspackings. Con- the characteristictime tv is comparableto with viscous liquids. Figure 2 shows the ex- sequerntly,the -activationbarrier that must mean relaxationtime TVwhen T is near Tg. tension of fliq(T) into the T < Tm regime of be surmounted,and the final change in The 0 = 1 limit in Eq. 5 correspondsto supercooling, under the assumption that basindepth will also only be 0(1). Because simple Debye relaxation,with tv servingas crystal nucleation has been avoided. In this the total kinetic energy is much larger the single structuralrelaxation time. How- extension, the system's configuration point (roughlyNkBT), enough thermalenergy is ever, smaller0 valueslead to a broaddistri- R(t) continues to wander as time t progresses virtually always available in principle to bution of relaxation times, and by trans- among the basins for amorphous inherent surmountthe interveningbarrier. The bot- forming gv to the frequencydomain this structures, without discovering entrance tleneck is that this kinetic energyis distrib- breadthbecomes explicit (12). Peaksin the channels to any of the absolute-minimum uted throughoutthe many-particlesystem, frequency-dependentabsorption then cor- basins of the crystalline state. Upon cooling and it maytake a verylong time for a proper respondapproximately to the dominantre- to absolute zero, the system becomes trapped fluctuation to concentrateenough kinetic laxation times. almost at random in one of these inherent- energyat the requiredlocation to effect the As Fig. 3 illustrates,the temperaturede- structure basiris of the amorphous state, fl,q transitionbetween basins. pendence of peak relaxationfrequency for becomes small (vanishing for classical statis- Relaxation response functions in the liquidsoften exhibits a bifurcation(13). In tics), and the system becomes a rigid glass. time domain, gj(t), to any of a variety of the equilibriumliquid range and into the weakexternal perturbations v (mechanical, moderatelysupercooled regime, there is a Arrested Kinetics electrical, thermal, optical, and so forth) single absorption maximum frequency. providean indicationof the actualrestruc- Upon approachingTg, this peaksplits into a The entire collection of fl(N) basins can be turing kinetics resulting from interbasin pairof maxima,the slow cx("primary") and classified by their depths. Let - /N transitions.If the assumptionthat the ini- faster f ("secondary")relaxations. The denote the inherent-structure potential en- tial-time normalizationgv(O) = 1 is im- former are non-Arrheniusand kinetically ergy for any given basin on a per-particle posed,the areasunder the gv(t) curvealong frozen out at Tg, and the latter are more basis. In analogy to Eq. 1 above, it can be the positive time axis define mean relax- nearlyArrhenius and remain operative at Tg. shown that the distribution of basins by ation times Tv(T).These dependsomewhat The a, L relaxation bifurcationhas a depth has the form N!exp[u(?)N] in the on property v, but in supercooled liquids straightforwardinterpretation in terms of large-N limit (5). A mean vibrational free they tend to display essentially a common the F) topography. As T declines, the con- energy per particle, fv can then be defined rapid rise with declining temperature and figuration point R(t) is forced into regions for basins of depth ?. The equilibrium state can often be fitted to a Vogel-Tammann- of increasingly rugged and heterogeneous

SCIENCE * VOL. 267 * 31 MARCH 1995 1937 topographyin order to seek out the ever deeperbasins that are identifiedby 4*q(T). The lower the temperature,the rarerand morewidely separated these basinsmust be. Leq However, the elementary transition processes that connect contiguousbasins con- tinue to requireonly local rearrangements of smallnumbers of particles.Evidently, the basinsare geometricallyorganized to create a two-scalelength and potentialenergy pat- Particle coordinates-, tern; Fig. 4 illustratesthis feature. Fig. 4. Two-scale potential energy topography The 1 processesare identifiedwith the characteristic of regions of configuration space elementaryrelaxations between neighbor- explored by fragile glass formers near Tg. The el- ing basins, whereasthe ot processesentail ementary interbasin transitions are associated escapefrom one deep basin within a large- with P relaxations, and large distance intercrater scale "crater"or "metabasin"and eventual- transitions are associated with ot relaxations. ly into another (14). Because the latter requiresa lengthy directedsequence of ele-- present. It is no surprise then that the ot, 0 0 TKTKTg~ T ~TM mentary transitions,it will acquire a net bifurcation is weak or absent in the strong Temperature elevation change (activationenergy) many cases. In contrast, the most fragile glass Fig. 5. Typical heat capacity curves for fragile formers indeed exhibit times that of the former. Also, the vast significant (D-scape glass formers in crystal (C), supercooled liquid intervening stretch of higher lying basins and distinctive ot, bifurcation. cratering (LsC),equilibrium liquid (Leq), and glass (G) phases; producesa large activationentropy. The relatively large effective singularity TK is the Kauzmann temperature. temperature To for fragile materials reflects Viscosity and Self-Diffusion the larger,and wider net barriers that they must surmount, as T declines toward To, for pared to the surroundings. Because such The relaxationalresponse of a fluid to shear the system configuration to pass from the domains are expected to be large on the stress is often describedby a specific in- interior of one inhabited crater to another of molecular scale near T and to have long stanceof Eq.4, the Maxwellrelaxation time comparable or greater depth. lifetimes, translational diffusion receives a TM(T),defined by the ratio of the shear The self-diffusion constant D measures disproportionate enhancement compared to viscosity q(T) to the elastic modulus G, the rate of increase with time, because of rotational diffusion, and a detailed analysis which is nearly temperatureindependent. Brownian motion, of the mean square dis- (18)-shows that the latter continues to ad- Consequently,the VTF formin Eq. 4 is also placement of a tagged particle in the medi- here more closely than the former to the a usefulrepresentation of shearviscosity [in- um. The Stokes-Einstein relation connects temperature dependence of -q(T). deed, this was its originalapplication (10)]: D to -q and to an effective hydrodynamic radius b for the particle: Ideal Glass State? Mq(T)- No exp[B/(T - To)] (6) = with virtuallythe same divergencetemper- D(T) kBT/[6Trb(T)] (7) Figure 5 illustrates schematically variation ature To that obtains for other types of (the constant 6,u. assumes a "sticking" with temperature of the heat capacity (Cp, relaxation.Experimentally, Tg's often corre- boundary condition at the particle surface). constant pressure) typically observed for spond to q in the range from 1011 to 1013 This relation has been remarkably success- fragile glass formers.Notable features are: (i) poise for nonpolymericliquids. ful at correlating independent measure- Cp rises discontinuously when the crystal The VTF representationfor viscosity ments of D(T) and si(T) for many liquids as melts at Tm; (ii) supercooling the liquid generates a useful classification of glass- both vary over many orders of magnitude in down toward Tg increases the discrepancy formingliquids between "strong" and "frag- the stable and supercooled liquid regimes. between Cp (liquid) and Cp (crystal); and ile" extremes(15), dependingon the value However, some recent experiments on frag- (iii) further cooling produces a nearly dis- of the ratio TO/Tm(or alternativelyTO/T). ile glass formers near Ta show a striking continuous drop in Cp (liquid) so that Cp The stronglimit has TO/Tm- 0, and q(t) breakdownplancemnwofl of aptggdpeartialcaleEq. 7; in these cases sthecmedi- D(T) (glass) is very close to that of Cp (crystal) at displays Arrhenius temperaturebehavior; becomes two orders of magnitude or more the same low temperature. In view of the Dxtoqatind toninanon efcive hydirodynamic real material examples that appear to be than the measured (T) would indi- basin-trapping interpretation of the glass rdomiusbnolarger thatencmassmn particle:, close to this limit are the oxide glassesSiO2 cate (16). Paradoxically, this dramatic ef- transition, it can be concluded that the in- and GeO2. The fragile limit displays dra- fect occurs while the corresponding rota- trabasin vibrational properties are Iargely matic non-Arrhenius q(T), and is illustrat- tionalboundrariy diffusionconiditionga rate is still theartil linked towr surfaince). (T) or the same for all basins, whether correspond- ed by orthoterphenyl,and the ionic mate- only weakly decoupled (17). ing to crystalline or to amorphous inherent rial KO.6ca0.4(NO3)1.4' Once again, the (D-scape cratering char- structures. Furthermore, most of the en- The variationin behaviorbetween the acteristic for fragile glass formers offers an hancement of Cp (liquid) over Cp (crystal) strong and fragile extremes can be traced explanation. When the temperature is low above Tg stems from the temperature varia- back to topographic differences in the and the system configuration point emerges tion of 4q,, the depth of the basins predom- 1D-scapesfor the respectivematerials. The from one deep crater to search for another, inately inhabited by the supercooled liquid. extremelimit of strongglass formers presents it has been emphasized that a long sequence The latent heat of melting causes the a uniformlyrough (single energyscale) to- of interbasin transitions will be liquid at Tm to possess a substantially higher elementary of m s pography, in which only the 13transitions of in Inmotio th is- entropy than the crystal. However, the C Fig. 4 have relevance. Little or no coherent discrepancy illustrated in Fig. 5 means that organization of the individual basins into the liquid loses entropy faster than the crys- large and deep craters, associated with the tal as both are cooled below Tm*The strong- low-temperature a~transitions, appears to be ly supercooled liquid at Tgstill has the higher

1938 SCIENCE * VOL. 267 * 31 MARCH 1995 FRONTIERS IN MATERIALS SCIENCE: ARTICLES

entropy,although the differencehas been tion has the furtherbenefit of uncovering 6. F. A. Lindemann,Z. Phys. 11, 609 (1910). 7. H. Lowen, Phys. Rep. 237, 249 (1994). substantiallyreduced. Smooth extrapolation and promotingseveral basic researchtopics 8. R. A. LaVioletteand F. H. Stillinger,J. Chem. Phys. of Cp (liquid) below Tg and of the corre- that need sustainedexperimental and the- 83, 4079 (1985). spondingentropy indicate the existenceof a oretical-simulationalattention. Examples of 9. F. H. Stillinger,ibid. 88, 7818 (1988). positive "Kauzmann temperature" TK at 10. G. W. Scherer, J. Am. Ceram. Soc. 75,1060 (1992). the latter are the material-specificenumer- 11. K. L. Ngai, Comments Solid State Phys. 9, 127 and which the crystal and the extrapolated liquid ation of inherent structures(FD minima), 141 (1979); R. Bohmer, K. L. Ngai, C. A. Angell, D. J. attain equal entropies (19). Considering that the exploitation of the Lindemann-like Plazek, J. Chem. Phys. 99, 4201 (1993). vibrational entropies are nearly the same for freezingcriterion for liquidsand its relation 12. E. W. Montrolland J. T. Bendler, J. Stat. Phys. 34, 129 (1984). the two phases, and that the inherent struc- to the glasstransition, and the criticaleval- 13. G. P. Johari and M. Goldstein, J. Chem. Phys. 53, tural entropy of the ordered crystal vanishes, uation of the ideal-glass-stateconcept. 2372 (1970). the fully relaxed glass at TK (the extrapolat- 14. F. H. Stillinger,Phys. Rev. B 41, 2409 (1990). 15. C. A. Angell, in Relaxations in Complex Systems, K. ed liquid) must also have vanishing inherent REFERENCESAND NOTES Ngai and G. B. Wright, Eds. (NationalTechnical In- structural entropy. This realization, coupled formation Service, U.S. Department of Commerce, 1. J. D. Bernal, Nature 183, 141 (1959); H. Eyringeta/., with the empirical observation that TK- Springfield, VA, 1985), p. 1. Proc. Natl. Acad. Sci. U.S.A. 44, 683 (1958); M. R. 16. F. Fujara,B. Geil, H. Sillescu, G. Fleischer, Z. Phys. B To, the mean relaxation-time divergence Hoare, Adv. Chem. Phys. 40, 49 (1979). 88, 195 (1992); M. T. Cicerone and M. D. Ediger, J. temperature,has generated the concept of an 2. M. Goldstein, J. Chem. Phys. 51, 3728 (1969); ibid. Phys. Chem. 97,10489 (1993); R. Kindet al., Phys. "ideal glass state" that could be experimen- 67, 2246 (1977). Rev. B 45, 7697 (1992). 3. For constant-pressure circumstances, system vol- 17. M. T. Cicerone, F. R. Blackburn, M. D. Ediger, J. tally attained if only sufficiently slow cooling ume becomes an additional coordinate and (D in- Chem. Phys. 102, 471 (1995). rates were available (20). cludes a pressure-volume contribution PV. 18. F. H. Stillingerand J. A. Hodgdon, Phys. Rev. E 50, If indeed it exists, the ideal glass state 4. F. H. Stillinger and T. A. Weber, Science 225, 983 2064 (1994). must correspond to the inherent structure (1984). 19. W. Kauzmann, Chem. Rev. 43, 219 (1948). 5. , Phys. Rev. A 25, 978 (1982). 20. J. Jackle, Rep. Prog. Phys. 49,171 (1986). with the lowest potential energy (deepest "crater") that is devoid of substantial regions with local crystalline order. Unfortu- nately, the details of this noncrystallinity constraint are unclear but may be crucial: The Microscopic Basis of the Qualifying inherent structures may depend on the maximum size and degree of perfec- Glass Transition in Polymers tion permitted crystalline inclusions in oth- erwise amorphous structures. This ambiq- uity, or non-uniqueness of choice criterion, from Neutron Scattering Studies would seem to undermine the concept of a substantially unique ideal glass state. B. Frick*and D. Richter It has also been argued (9) that the seemingly innocuous extrapolations that Recent neutron scattering experiments on the microscopic dynamics of polymers below identify a positive Kauzmann temperature and above the glass transition temperature T1 are reviewed. The results presented cover TK (and by implication To) are flawed. different dynamic processes appearing in glasses: local motions, vibrations, and different Localized particle rearrangements (associat- relaxation processes such as (- and f-relaxation. For the a-relaxation, which occurs ed with I relaxations) are always possible, above Tg, it is possible to extend the time-temperature superposition principle, which is even in a hypothetical ideal glass structure, valid for polymers on a macroscopic scale, to the microscopic time scale. However, this and raise the potential energy only by 0(1). principle is not applicable for temperatures approaching Tg. Below Tg, an inelastic exci- These structural excitations in the strict tation at a frequency of some hundred gigahertz (on the order of several wave numbers), sense prevent attaining the ideal glass state the "boson peak," survives from a quasi-elastic overdamped scattering law at high at positive temperature, in conflict with the temperatures. The connection between this boson peak and the fast dynamic process usual view (20). appearing near Tg is discussed. In spite of these formal reservations, the ideal glass state concept remains valuable. Careful and systematic experiments on the most fragile glass formers should help to Polymershave a verywide range of appli- solid and the viscous liquid are better un- remove some of the obscuring uncertainties. cability.and can be 'usedin the solid, rub- derstood than the viscoelastic regime, al- bery, or molten states. Solid polymersare thougha largeamount of experimentaldata Conclusions used whenever elastic strengthis required, existsfor the viscoelasticstate (1). Polymers and melts are used wheneverviscous prop- are generallyamorphous solids, that is, they By focusing attention on the topographic ertiesare desired. The intermediaterange of are microscopically disordered without characteristics of (D in the multidimension- viscoelasticity has particularlyinteresting translationalsymmetry and crystallizationis al configuration space, a comprehensive de- propertiesand covers an especially wide rare.The microscopicdisorder remains es- scription becomes available for static and temperaturerange for polymersbecause of sentially unchangedas the polymertrans- kinetic phenomena exhibited by super- their chain structure.Ideally, an under- formsfrom the glassysolid state to the melt cooled liquids and their glass transitions. In standing of the microscopic dynamics or liquid state (Fig. IA). particular, this viewpoint rationalizes the wouldbe a prerequisitefor optimizedappli- The key to understandingthe property characteristic properties of fragile glass cation. The limiting cases of the elastic changes is the change of the microscopic formers, including non-Arrhenius viscosity, dynamics,and any structuralchanges are B. Frickis in the InstitutLaue-Langevin, BP1 56, F-38042 primary-secondary relaxation bifurcation, Grenoble Cedex 9, France. D. Richter is in the Institutfur thereforea consequenceof it. Microscopi- and the enhancement of self-diffusion rates Festkorperforschung, Forschungsanlage Julich, D-52425 cally,elastic solids are characterized by their over the Stokes-Einstein prediction. This Julich, Germany. atoms or molecules being bound within a multidimensional topographic representa- *To whom correspondence should be addressed. potentialdefined by the surroundingatoms.

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