Relations between a Liquid and Its Glasses

Celebrating Soft Matter's 10th anniversary: Testing the foundations of classical entropy: colloid experiments

Defining the entropy of classical particles raises a number of paradoxes and ambiguities, some of which have been known for over a century. Several, such as Gibbs' paradox, involve the fact that classical particles are distinguishable, and in textbooks these are often 'resolved' by appeal to the quantum-mechanical indistinguishability of atoms or molecules of the same type. However, questions then remain of how to correctly define the entropy of large poly-atomic particles such as colloids in suspension, of which no two are exactly alike. By performing experiments on such colloids, one can establish that certain definitions of the classical entropy fit the data, while others in the literature do not. Specifically, the experimental facts point firmly to an 'informatic' interpretation that dates back to Gibbs: entropy is determined by the number of microstates that we as observers choose to treat as equivalent when we identify a macrostate. This approach, unlike some others, can account for the existence of colloidal crystals, and for the observed abundances of colloidal clusters of different shapes. We also address some lesser-known paradoxes whereby the physics of colloidal assemblies, which ought to be purely classical, seems to involve quantum mechanics directly. The experimental symptoms of such involvement are predicted to be 'isotope effects' in which colloids with different inertial masses, but otherwise identical sizes and properties, show different aggregation statistics. These paradoxes are caused by focussing one's attention on some classical degrees while neglecting others; when all are treated equally, all isotope effects are found to vanish.

Computer simulations of glasses: the potential energy landscape

We review the current state of research on glasses, discussing the theoretical background and computational models employed to describe them. This article focuses on the use of the potential energy landscape (PEL) paradigm to account for the phenomenology of glassy systems, and the way in which it can be applied in simulations and the interpretation of their results. This article provides a broad overview of the rich phenomenology of glasses, followed by a summary of the theoretical frameworks developed to describe this phenomonology. We discuss the background of the PEL in detail, the onerous task of how to generate computer models of glasses, various methods of analysing numerical simulations, and the literature on the most commonly used model systems. Finally, we tackle the problem of how to distinguish a good glass former from a good crystal former from an analysis of the PEL. In summarising the state of the potential energy landscape picture, we develop the foundations for new theoretical methods that allow the ab initio prediction of the glass-forming ability of new materials by analysis of the PEL.

Extension of the BMCSL equation of state for hard spheres to the metastable disordered region: Application to the SAFT approach

A simple modification of the Boublík-Mansoori-Carnahan-Starling-Leland equation of state is proposed for an application to the metastable disordered region. The new model has a positive pole at the jamming limit and can accurately describe the molecular simulation data of pure hard in the stable fluid region and along the metastable branch. The new model has also been applied to binary mixtures hard spheres, and an excellent description of the fluid and metastable branches can be obtained by adjusting the jamming packing fraction. The new model for hard sphere mixtures can be used as the repulsive term of equations of state for real fluids. In this case, the modified equations of state give very similar predictions of thermodynamic properties as the original models, and one can remove the multiple liquid density roots observed for some versions of the Statistical Associating Fluid Theory (SAFT) at low temperature without any modification of the dispersion term.

Dynamics and thermodynamics of polymer glasses

The fate of matter when decreasing the temperature at constant pressure is that of passing from gas to liquid and, subsequently, from liquid to crystal. However, a class of materials can exist in an amorphous phase below the melting temperature. On cooling such materials, a glass is formed; that is, a material with the rigidity of a solid but exhibiting no long-range order. The study of the thermodynamics and dynamics of glass-forming systems is the subject of continuous research. Within the wide variety of glass formers, an important sub-class is represented by glass forming polymers. The presence of chain connectivity and, in some cases, conformational disorder are unfavourable factors from the point of view of crystallization. Furthermore, many of them, such as amorphous thermoplastics, thermosets and rubbers, are widely employed in many applications. In this review, the peculiarities of the thermodynamics and dynamics of glass-forming polymers are discussed, with particular emphasis on those topics currently the subject of debate. In particular, the following aspects will be reviewed in the present work: (i) the connection between the pronounced slowing down of glassy dynamics on cooling towards the glass transition temperature (Tg) and the thermodynamics; and, (ii) the fate of the dynamics and thermodynamics below Tg. Both aspects are reviewed in light of the possible presence of a singularity at a finite temperature with diverging relaxation time and zero configurational entropy. In this context, the specificity of glass-forming polymers is emphasized.

Fragile-strong fluid crossover and universal relaxation times in a confined hard-disk fluid

We show that a system of hard disks confined to a narrow channel exhibits a fragile-strong fluid crossover located at the maximum of the isobaric heat capacity and that the relaxation times for different channel widths fall onto a single master curve when rescaled by the relaxation times and temperatures of the crossover. Calculations of the configurational entropy and the inherent structure equation of state find that the crossover is related to properties of the jamming landscape for the model but that the Adam-Gibbs relation does not predict the relaxation behavior. We also show that a facilitated dynamics description of the system, where kinetically excited regions are identified with local packing arrangements of the disks, successfully describes the fragile-strong crossover.

Predicted Optimum Composition for the Glass-Forming Ability of Bulk Amorphous Alloys: Application to Cu-Zr-Al

Metallic glasses have been established to have unique properties such as ductility, toughness, and soft magnetism with promising engineering applications. However, the glass-forming ability (GFA) has not been sufficient to synthesize the bulk metallic glasses (BMGs) required for many engineering applications. Attempts to develop the understanding of the GFA required to predict the optimum alloys have not yet been proven successful. We develop here a computational model based on molecular dynamics simulations that explains the dramatic change of GFA with alloying small amounts of Al into Cu-Zr. We find that the high GFA to form BMGs depends on a combination of three factors, (a) a low thermodynamic driving force for crystallization, (b) a high melt viscosity, and (c) large ratios of icosahedral structures in the liquid phase. These computational methods to predict these factors that suppress formation of crystal nuclei and slow the dynamic motions in the liquids are practical for in silico prediction of new alloys with optimal GFA.

Dependence of the fragility of a glass former on the softness of interparticle interactions

We study the influence of the softness of the interparticle interactions on the fragility of a glass former by considering three model binary mixture glass formers. The interaction potential between particles is a modified Lennard-Jones type potential, with the repulsive part of the potential varying with an inverse power q of the interparticle distance, and the attractive part varying with an inverse power p. We consider the combinations (12,11) (model I), (12,6) (model II), and (8,5) (model III) for (q,p) such that the interaction potential becomes softer from model I to III. We evaluate the kinetic fragilities from the temperature variation of diffusion coefficients and relaxation times, and a thermodynamic fragility from the temperature variation of the configurational entropy. We find that the kinetic fragility increases with increasing softness of the potential, consistent with previous results for these model systems, but at variance with the thermodynamic fragility, which decreases with increasing softness of the interactions, as well as expectations from earlier results. We rationalize our results by considering the full form of the Adam-Gibbs relation, which requires, in addition to the temperature dependence of the configurational entropy, knowledge of the high temperature activation energies in order to determine fragility. We show that consideration of the scaling of the high temperature activation energy with the liquid density, analyzed in recent studies, provides a partial rationalization of the observed behavior.

Raman scattering boson peak and differential scanning calorimetry studies of the glass transition in tellurium-zinc oxide glasses

Raman scattering and differential scanning calorimetry (DSC) measurements have been carried out on four mixed tellurium-zinc oxide (TeO(2))(1 - x)(ZnO)(x) (x = 0.1, 0.2, 0.3, 0.4) glasses under variable temperature, with particular attention being given to the respective glass transition region. From the DSC measurements, the glass transition temperature T(g) has been determined for each glass, showing a monotonous decrease of T(g) with increasing ZnO content. The Raman study is focused on the low-frequency band of the glasses, the so-called boson peak (BP), whose frequency undergoes an abrupt decrease at a temperature T(d) very close to the respective T(g) values obtained by DSC. These results show that the BP is highly sensitive to dynamical effects over the glass transition and provides a means for an equally reliable (to DSC) determination of T(g) in tellurite glasses and other network glasses. The discontinuous temperature dependence of the BP frequency at the glass transition, along with the absence of such a behaviour by the high-frequency Raman bands (due to local atomic vibrations), indicates that marked changes of the medium range order (MRO) occur at T(g) and confirms the correlation between the BP and the MRO of glasses.

Understanding fragility in supercooled Lennard-Jones mixtures. II. Potential energy surface

The connection between isobaric fragility and the properties of high-order stationary points of the potential energy surface in different supercooled Lennard-Jones mixtures was investigated. The increase of effective activation energies upon supercooling appears to be driven by the increase of average potential energy barriers measured by the energy dependence of the fraction of unstable modes. Such an increase is sharper, the more fragile the mixture. Correlations between fragility and other properties of high-order stationary points, including the vibrational density of states and the localization features of unstable modes, are also discussed.

Single-order-parameter description of glass-forming liquids: A one-frequency test

Thermoviscoelastic linear-response functions are calculated from the master equation describing viscous liquid inherent dynamics. From the imaginary parts of the frequency-dependent isobaric specific heat, isothermal compressibility, and isobaric thermal expansion coefficient, we define a "linear dynamic Prigogine-Defay ratio" LambdaTp(omega) with the property that if LambdaTp(omega)=1 at one frequency, then LambdaTp(omega) is unity at all frequencies. This happens if and only if there is a single-order-parameter description of the thermoviscoelastic linear responses via an order parameter (which may be nonexponential in time). Generalizations to other cases of thermodynamic control parameters than temperature and pressure are also presented.

Thermodynamic Basis for Cluster Kinetics: Prediction of the Fragility of Marginal Metallic Glass-Forming Liquids

Due to the inaccessibility of the supercooled region of marginal metallic glasses (MMGs) within the experimental time window, we study the cluster kinetics above the liquidus temperature, Tl, to acquire information on the fragility of the MMG systems. The thermodynamic basis for the stability of locally ordered structure in the MMG liquids is discussed in terms of the two-order-parameter model. It is found that the Arrhenius activation energy of clusters, Deltah, is proportional to the chemical mixing enthalpy of alloys, DeltaH(chem). Fragility of the MMG forming liquids can be described by the ratio of the absolute DeltaH(chem) value to the glass transition temperature, Tg. The manner of vitrification during rapid solidification is an important factor for the discrepancy between the data presented in this paper and the prediction of the two-order-parameter model concerning the relation between Delta h and the liquid fragility.

A thermodynamic approach to the fragility of glass-forming polymers

We have connected the dynamic fragility, namely, the steepness of the relaxation-time variation upon temperature reduction, to the excess entropy and heat capacity of a large number of glass-forming polymers. The connection was obtained in a natural way from the Adam-Gibbs equation, relating the structural relaxation time to the configurational entropy. We find a clear correlation for a group of polymers. For another group of polymers, for which this correlation does not work, we emphasize the role of relaxation processes unrelated to the alpha process in affecting macroscopic thermodynamic properties. Once the residual excess entropy at the Vogel temperature is removed from the total excess entropy, the correlation between dynamic fragility and thermodynamic properties is reestablished.

Spatially Heterogeneous Dynamics and the Adam−Gibbs Relation in the Dzugutov Liquid

We perform molecular dynamics simulations of a one-component glass-forming liquid and use the inherent structure formalism to test the predictions of the Adam-Gibbs (AG) theory and to explore the possible connection between these predictions and spatially heterogeneous dynamics. We calculate the temperature dependence of the average potential energy of the equilibrium liquid and show that it obeys the Rosenfeld-Tarazona T(3/5) law for low temperature T, while the average inherent structure energy is found to be inversely proportional to temperature at low T, consistent with a Gaussian distribution of potential energy minima. We investigate the shape of the basins around the local minima in configuration space via the average basin vibrational frequency and show that the basins become slightly broader upon cooling. We evaluate the configurational entropy S(conf), a measure of the multiplicity of potential energy minima sampled by the system, and test the validity of the AG relation between S(conf) and the bulk dynamics. We quantify the dynamically heterogeneous motion by analyzing the motion of particles that are mobile on short and intermediate time scales relative to the characteristic bulk relaxation time. These mobile particles move in one-dimensional "strings", and these strings form clusters with a well-defined average cluster size. The AG approach predicts that the minimum size of cooperatively rearranging regions (CRR) of molecules is inversely proportional to S(conf), and recently (Phys. Rev. Lett. 2003, 90, 085506) it has been shown that the mobile-particle clusters are consistent with the CRR envisaged by Adam and Gibbs. We test the possibility that the mobile-particle strings, rather than clusters, may describe the CRR of the Adam-Gibbs approach. We find that the strings also follow a nearly inverse relation with S(conf). We further show that the logarithm of the time when the strings and clusters are maximum, which occurs in the late-beta-relaxation regime of the intermediate scattering function, follows a linear relationship with 1/TS(conf), in agreement with the AG prediction for the relationship between the configurational entropy and the characteristic time for the liquid to undergo a transition to a new configuration. Since strings are the basic elements of the clusters, we propose that strings are a more appropriate measure of the minimum size of a CRR that leads to configurational transitions. That the cluster size also has an inverse relationship with S(conf) may be a consequence of the fact that the clusters are composed of strings.

Configurational entropy and its crisis in metastable states: ideal glass transition in a dimer model as a paragidm of a molecular glass

We discuss the need for discretization to evaluate the configurational entropy in a general model. We also discuss the prescription using restricted partition function formalism to study the stationary limit of metastable states where a more stable equilibrium state exists. We introduce a lattice model of dimers as a paradigm of molecular fluid and study stationary metastability in it to investigate the root cause of glassy behavior. We demonstrate the existence of entropy crisis in metastable states, from which it follows that the entropy crisis is the root cause underlying the ideal glass transition in systems with particles of all sizes. The orientational interactions in the model control the nature of the liquid-liquid transition observed in recent years in molecular glasses.

Thermodynamics and Dynamics of Metallic Glass Formers: Their Correlation for the Investigation on Potential Energy Landscape

Great progress has been made in basic features of the potential energy landscape (PEL) theoretically. The present work, however, attempts to cast new light on it from experimental aspects. By a survey of experimental data related to thermodynamics or dynamics of metallic glass-forming liquids, it is found that the increased rate of excitation of vibrational entropy at glass transition tends to increase the rate of generation of configurational part. Although for the type of metallic materials a generally positive relationship exists between the density of the energy minima at glass transition and the liquid fragility strength, just as expected, our main attention is paid to the phenomenon of the scattering of the slopes. Analysis shows that the phenomenon results from the different average height of energy barriers between minima near glass transition. Investigation on the PdNiP metallic system indicates that the mismatch entropy is a dominant factor in the barrier height: a large value of it results in low energy barriers. Our previous work on the AlNiCe system gives the support to this finding.

Effect of chemical structure on the isobaric and isochoric fragility in polychlorinated biphenyls

Pressure-volume-temperature data, along with dielectric relaxation measurements, are reported for a series of polychlorinated biphenyls (PCB), differing in the number of chlorine atoms on their phenyl rings. Analysis of the results reveals that with increasing chlorine content, the relaxation times of the PCB become governed to a greater degree by density rho relative to the effect of temperature T. This result is consistent with the respective magnitudes of the scaling exponent gamma yielding superpositioning of the relaxation times measured at various temperatures and pressures, when plotted versus rho(gamma)/T. While at constant (atmospheric) pressure, fragilities for the various PCB are equivalent, the fragility at constant volume varies inversely with chlorine content. Evidently, the presence of bulkier chlorine atoms on the phenyl rings magnifies the effect which the density has on the relaxation dynamics.

Distinct Difference in Ionic Transport Behavior in Polymer Electrolytes Depending on the Matrix Polymers and Incorporated Salts

Two different electrolyte salts, lithium bis(trifluoromethanesulfonyl)imide (LiTFSI), and a room temperature ionic liquid, 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide (EMITFSI), were incorporated into network polymers to obtain ion-conductive polymer electrolytes. Network polymers of poly(ethylene oxide-co-propylene oxide) (P(EO/PO)) and poly(methyl methacrylate) (PMMA) were chosen as matrixes for LiTFSI and EMITFSI, respectively. Both of the polymer electrolytes were single-phase materials and were completely amorphous. Ionic conductivity of the polymer electrolytes was measured over a wide temperature range, with the lowest temperatures close to or below the glass transition temperatures (Tg). The Arrhenius plots of the conductivity for both of the systems exhibited positively curved profiles and could be well fit to the Vogel-Tamman-Fulcher (VTF) equation. The conductivity of the PMMA/EMITFSI electrolytes was higher at most by 3 orders of magnitude than that of the LiTFSI/P(EO/ PO) electrolytes at ambient temperature. When the ideal glass transition temperature, T0 (one of the VTF fitting parameters), was compared with the Tg, a difference in the ionic conduction was apparent in these systems. In the P(EO/PO)/LiTFSI electrolytes, the T0 and Tg increased in parallel with salt concentration and the T0 was lower than the Tg by ca. 50 degrees C. On the contrary, the difference between the T0 and the Tg increased with increasing content of PMMA in the PMMA/EMITFSI electrolytes, with the observed difference in the concentration range studied reaching up to ca. 100 degrees C. The conductivity at the Tg, sigma(Tg), for the LiTFSI/P(EO/PO) electrolytes was on the order of 10(-14-)10(-13) S cm(-1) and increased with increasing salt concentration, whereas that for the PMMA/EMITFSI polymer electrolytes reached 10(-7) S cm(-1) when the concentration of PMMA was high. The ion transport mechanism was discussed in terms of the concepts of coupling/decoupling and strong/fragile for the two different polymer electrolytes.

Does the interaction potential determine both the fragility of a liquid and the vibrational properties of its glassy state?

By performing molecular dynamics simulations of binary Lennard-Jones systems with three different potentials, we show that the increase of anharmonicity and capacity for intermolecular coupling of the potential is the cause of (i) the increase of kinetic fragility and nonexponentiality in the liquid state, and (ii) the T(g)-scaled temperature dependence of the nonergodicity parameter determined by the vibrations at low temperatures in the glassy state. Naturally, these parameters correlate with each other, as observed experimentally by T. Scopigno et al. [Science 302, 849 (2003)]

New phase for one-component hard spheres

A completely new phase for one-component hard spheres is reported in an unexpected region of the phase diagram. The new phase is observed at compressibility factors intermediate between the solid and the metastable branches. It can be obtained from either Monte Carlo simulations alone or a combination of Monte Carlo and molecular dynamics calculations. An analysis of the intermediate scattering function data shows that the new phase is in a stable equilibrium. Radial distribution function data, configurational snapshots, bond order parameters, and translational order parameters obtained from molecular simulations indicate that the new phase is significantly different from the isotropic liquid, metastable, or crystalline phases traditionally observed in hard sphere systems. This result significantly changes our previous understanding of the behavior of hard spheres.

Landscapes and fragilities

The concept of fragility provides a possibility to rank different supercooled liquids on the basis of the temperature dependence of dynamic and/or thermodynamic quantities. We recall here the definitions of kinetic and thermodynamic fragility proposed in the last years and discuss their interrelations. At the same time we analyze some recently introduced models for the statistical properties of the potential energy landscape. Building on the Adam-Gibbs relation, which connects structural relaxation times to configurational entropy, we analyze the relation between statistical properties of the landscape and fragility. We call attention to the fact that the knowledge of number, energy depth, and shape of the basins of the potential energy landscape may not be sufficient for predicting fragility. Finally, we discuss two different possibilities for generating strong behavior.

Fragility in p-spin models

We investigate the relation between fragility and phase space properties - such as the distribution of states - in the mean-field p -spin model, a solvable model that has been frequently used in studies of the glass transition. By direct computation of all the relevant quantities, we find that (i) the recently observed correlation between fragility and vibrational properties at low temperature is present in this model and (ii) the total number of states is a decreasing function of fragility, at variance with what is currently believed. We explain these findings by taking into account the contribution to fragility coming from the transition paths between different states. Finally, we propose a geometric picture of the phase space that explains the correlation between properties of the transition paths, distribution of states, and their vibrational properties. However, our analysis may not apply to strong systems where inflection points in the configurational entropy as a function of the temperature are found.

Spinodal and ideal glass limits for metastable liquids

A spinodal line provides a high-temperature limit to the stability of a superheated liquid and a line of ideal glass transitions may provide a low-temperature limit for a supercooled liquid. For models in which the thermodynamic properties depend on only one independent external variable the line of ideal glass transition can be viewed as a second-order transition between two equilibrated phases and it meets the spinodal with the same slope at the maximum tension a stretched liquid can sustain. For real materials there are two independent external variables, temperature and pressure; the line of ideal glass transitions cannot be viewed as a second-order phase transition and the two lines need not have the same slope where they meet.

Thermodynamics and the glass transition in model energy landscapes

We determine the liquid-state thermodynamics for a model energy landscape corresponding to soft spheres with a mean-field attraction. We consider two approximations, in which the distribution of potential energy minima is either Gaussian or binomial, and for each we calculate the liquid spinodal, binodal, and "effective" glass transition locus. The resulting models provide a unified description of the liquid state across the complete range from low-temperature glassiness to high-temperature instability with respect to the vapor phase.

Is the Fragility of a Liquid Embedded in the Properties of Its Glass?

When a liquid is cooled below its melting temperature, it usually crystallizes. However, if the quenching rate is fast enough, the system may remain in a disordered state, progressively losing its fluidity upon further cooling. When the time needed for the rearrangement of the local atomic structure reaches approximately 100 seconds, the system becomes "solid" for any practical purpose, and this defines the glass transition temperature Tg. Approaching this transition from the liquid side, different systems show qualitatively different temperature dependencies of the viscosity, and accordingly they have been classified by introducing the concept of "fragility." We report experimental observations that relate the microscopic properties of the glassy phase to the fragility. We find that the vibrational properties of the glass well below Tg are correlated with the fragility value. Consequently, we extend the fragility concept to the glassy state and indicate how to determine the fragility uniquely from glass properties well below Tg.

Kauzmann's paradox and the glass transition

Kauzmann showed that the entropy of a liquid decreases rapidly on cooling towards the kinetic glass transition temperature and extrapolates to unreasonable values at lower temperature. The temperature where the extrapolated liquid entropy meets the crystal entropy is now called the Kauzmann temperature. Thermodynamics, with Planck's statement of the third law, shows that the entropy of a liquid cannot be less than the entropy of a glass with the same enthalpy. This is the thermodynamic condition violated by the Kauzmann extrapolation and it suggests a thermodynamic glass transition. Simulations show that, for the simple models studied and regardless of how the liquid entropy is extrapolated, the Kauzmann temperature cannot be reached because the entropy of glasses with the same enthalpy as the liquid is greater than that of the crystal.

Relation between thermodynamics and kinetics of glass-forming liquids

Vitrification of a supercooled liquid is often characterized by the hypothetical kinetic instability point, the Vogel-Fulcher temperature T0, and the thermodynamic one, the Kauzmann temperature T(K). The widely believed relation T0 congruent with T(K) is regarded as the supporting evidence of a direct connection between the thermodynamics and kinetics of glass-forming liquids. Here we demonstrate that T(K)/T(0) systematically increases from unity with a decrease in the fragility, contrary to the common belief. This systematic deviation may be explained by a synergistic effect between the weaker cooperativity and the stronger tendency of short-range ordering in stronger glass formers.

Potential energy landscape equation of state

Depth, number, and shape of the basins of the potential energy landscape are the key ingredients of the inherent structure thermodynamic formalism introduced by Stillinger and Weber [F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, 978 (1982)]. Within this formalism, an equation of state based only on the volume dependence of these landscape properties is derived. Vibrational and configurational contributions to pressure are sorted out in a transparent way. Predictions are successfully compared with data from extensive molecular dynamics simulations of a simple model for the fragile liquid orthoterphenyl.

Spatially correlated dynamics in a simulated glass-forming polymer melt: analysis of clustering phenomena

In recent years, experimental and computational studies have demonstrated that the dynamics of glass-forming liquids are spatially heterogeneous, exhibiting regions of temporarily enhanced or diminished mobility. Here we present a detailed analysis of dynamical heterogeneity in a simulated "bead-spring" model of a low-molecular-weight polymer melt. We investigate the transient nature and size distribution of clusters of "mobile" chain segments (monomers) as the polymer melt is cooled toward its glass transition. We also explore the dependence of this clustering on the way in which the mobile subset is defined. We show that the mean cluster size is time dependent with a peak at intermediate time, and that the mean cluster size at the peak time grows with decreasing temperature T. We show that for each T a particular fraction of particles maximizes the mean cluster size at some characteristic time, and this fraction depends on T. The growing size of the clusters demonstrates the growing range of correlated motion, previously reported for this same system [C. Beneman et al. Nature (London) 399, 246 (1999)]. The distribution of cluster sizes approaches a power law near the mode-coupling temperature, similar to behavior reported for a simulated binary mixture and a dense colloidal suspension, but with a different exponent. We calculate the correlation length of the clusters, and show that it exhibits similar temperature- and time-dependent behavior as the mean cluster size, with a maximum at intermediate time. We show that the characteristic time of the maximum cluster size follows the scaling predicted by mode-coupling theory (MCT) for the beta time scale, revealing a possible connection between spatially heterogeneous dynamics and MCT.

Fragile-to-strong transition and polyamorphism in the energy landscape of liquid silica

Liquid silica is the archetypal glass former, and compounds based on silica are ubiquitous as natural and man-made amorphous materials. Liquid silica is also the extreme case of a 'strong' liquid, in that the variation of viscosity with temperature closely follows the Arrhenius law as the liquid is cooled toward its glass transition temperature. In contrast, most liquids are to some degree 'fragile', showing significantly faster increases in their viscosity as the glass transition temperature is approached. Recent studies have demonstrated the controlling influence of the potential energy hypersurface (or 'energy landscape') of the liquid on the transport properties near the glass transition. But the origin of strong liquid behaviour in terms of the energy landscape has not yet been resolved. Here we study the static and dynamic properties of liquid silica over a wide range of temperature and density using computer simulations. The results reveal a change in the energy landscape with decreasing temperature, which underlies a transition from a fragile liquid at high temperature to a strong liquid at low temperature. We also show that a specific heat anomaly is associated with this fragile-to-strong transition, and suggest that this anomaly is related to the polyamorphic behaviour of amorphous solid silica.

A thermodynamic connection to the fragility of glass-forming liquids

Although liquids normally crystallize on cooling, there are members of all liquid types (including molecular, ionic and metallic) that supercool and then solidify at their glass transition temperature, Tg. This continuous solidification process exhibits great diversity within each class of liquid-both in the steepness of the viscosity-temperature profile, and in the rate at which the excess entropy of the liquid over the crystalline phase changes as Tg is approached. However, the source of the diversity is unknown. The viscosity and associated relaxation time behaviour have been classified between 'strong' and 'fragile' extremes, using Tg as a scaling parameter, but attempts to correlate such kinetic properties with the thermodynamic behaviour have been controversial. Here we show that the kinetic fragility can be correlated with a scaled quantity representing excess entropy, using data over the entire fragility range and embracing liquids of all classes. The excess entropy used in our correlation contains both configurational and vibration-related contributions. In order to reconcile our correlation with existing theory and simulations, we propose that variations in the fragility of liquids originate in differences between their vibrational heat capacities, harmonic and anharmonic, which we interpret in terms of an energy landscape. The differences evidently relate to behaviour of low-energy modes near and below the boson peak.

Thermodynamic and structural aspects of the potential energy surface of simulated water

Relations between the thermodynamics and dynamics of supercooled liquids approaching a glass transition is a topic of considerable interest. The potential energy surface of model liquids has been increasingly studied, since it provides a connection between the configurational component of the partition function on the one hand, and the system dynamics on the other. This connection is most obvious at low temperatures, where the motion of the system can be partitioned into vibrations within a basin of attraction and infrequent interbasin transitions. In this work, we present a description of the potential energy surface properties of supercooled liquid water. The dynamics of this model have been studied in great detail in recent years. We locate the minima sampled by the liquid by "quenches" from equilibrium configurations generated via molecular dynamics simulations, and then calculate the temperature and density dependence of the basin energy, degeneracy, and shape. The temperature dependence of the energy of the minima is qualitatively similar to simple liquids, but has anomalous density dependence. The unusual density dependence is also reflected in the configurational entropy, the thermodynamic measure of degeneracy. Finally, we study the structure of simulated water at the minima, which provides insight on the progressive tetrahedral ordering of the liquid on cooling.

Supercooled liquids and the glass transition

Glasses are disordered materials that lack the periodicity of crystals but behave mechanically like solids. The most common way of making a glass is by cooling a viscous liquid fast enough to avoid crystallization. Although this route to the vitreous state-supercooling-has been known for millennia, the molecular processes by which liquids acquire amorphous rigidity upon cooling are not fully understood. Here we discuss current theoretical knowledge of the manner in which intermolecular forces give rise to complex behaviour in supercooled liquids and glasses. An intriguing aspect of this behaviour is the apparent connection between dynamics and thermodynamics. The multidimensional potential energy surface as a function of particle coordinates (the energy landscape) offers a convenient viewpoint for the analysis and interpretation of supercooling and glass-formation phenomena. That much of this analysis is at present largely qualitative reflects the fact that precise computations of how viscous liquids sample their landscape have become possible only recently.