Heat capacity, enthalpy fluctuations, and configurational entropy in broken ergodic systems

Kauzmann Paradox, Supercooling, and Finding Order in Chaos

Prediction of a liquidus state with lower entropy than the corresponding solid at Kauzmann temperature (Tk), and associated entropy catastrophe/paradox, remains an enigma. Despite efforts to resolve this paradox for nearly 80 years, no unifying resolution has been reported. Potential resolutions to the Kauzmann paradox rely on an ideal glass transition, however, this limits the interpretation of Tk as an equilibrium critical point rather than an instability. Focusing on entropy, statistical mechanics and non-equilibrium dynamics becomes a key tenet in resolving this paradox. Expansion in phase space beyond 2D and consideration of Tk as a non-equilibrium critical point is necessary to understand the extent of liquid relaxation beyond Tk. In this review, we provide an entropic perspective of the relaxation behavior of supercooled liquids, associated expanded phase diagram, and the potential resolution to the Kauzmann paradox. This work integrates the historical evolution of our understanding of entropy/thermodynamics with modern interpretation of quantum states through renormalization group and thermodynamic speed limits.

Role of anisotropy in understanding the molecular grounds for density scaling in dynamics of glass-forming liquids

Molecular Dynamics (MD) simulations of glass-forming liquids play a pivotal role in uncovering the molecular nature of the liquid vitrification process. In particular, much focus was given to elucidating the interplay between the character of intermolecular potential and molecular dynamics behaviour. This has been tried to achieve by simulating the spherical particles interacting via isotropic potential. However, when simulation and experimental data are analysed in the same way by using the density scaling approaches, serious inconsistency is revealed between them. Similar scaling exponent values are determined by analysing the relaxation times and pVT data obtained from computer simulations. In contrast, these values differ significantly when the same analysis is carried out in the case of experimental data. As discussed thoroughly herein, the coherence between results of simulation and experiment can be achieved if anisotropy of intermolecular interactions is introduced to MD simulations. In practice, it has been realized in two different ways: (1) by using the anisotropic potential of the Gay-Berne type or (2) by replacing the spherical particles with quasi-real polyatomic anisotropic molecules interacting through isotropic Lenard-Jones potential. In particular, the last strategy has the potential to be used to explore the relationship between molecular architecture and molecular dynamics behaviour. Finally, we hope that the results presented in this review will also encourage others to explore how 'anisotropy' affects remaining aspects related to liquid-glass transition, like heterogeneity, glass transition temperature, glass forming ability, etc.

Melt-quenched carboxylate metal-organic framework glasses

Although carboxylate-based frameworks are commonly used architectures in metal-organic frameworks (MOFs), liquid/glass MOFs have thus far mainly been obtained from azole- or weakly coordinating ligand-based frameworks. This is because strong coordination bonds of carboxylate ligands to metals block the thermal vitrification pathways of carboxylate-based MOFs. In this study, we present the example of carboxylate-based melt-quenched MOF glasses comprising Mg2+ or Mn2+ with an aliphatic carboxylate ligand, adipate. These MOFs have a low melting temperature (Tm) of 284 °C and 238 °C, respectively, compared to zeolitic-imidazolate framework (ZIF) glasses, and superior mechanical properties in terms of hardness and elastic modulus. The low Tm may be attributed to the flexibility and low symmetry of the aliphatic carboxylate ligand, which raises the entropy of fusion (ΔSfus), and the lack of crystal field stabilization energy on metal ions, reducing enthalpy of fusion (ΔHfus). This research will serve as a cornerstone for the integration of numerous carboxylate-based MOFs into MOF glasses.

Emerging exotic compositional order on approaching low-temperature equilibrium glasses

The ultimate fate of a glass former upon cooling has been a fundamental problem in condensed matter physics and materials science since Kauzmann. Recently, this problem has been challenged by a model with an extraordinary glass-forming ability effectively free from crystallisation and phase separation, two well-known fates of most glass formers, combined with a particle-size swap method. Thus, this system is expected to approach the ideal glass state if it exists. However, we discover exotic compositional order as the coexistence of space-spanning network-like structures formed by small-large particle connections and patches formed by medium-size particles at low temperatures. Therefore, the glass transition is accompanied unexpectedly by exotic compositional ordering inaccessible through ordinary structural or thermodynamic characterisations. Such exotic compositional ordering is found to have an unusual impact on structural relaxation dynamics. Our study thus raises fundamental questions concerning the role of unconventional structural ordering in understanding glass transition.

Entropy Scaling of Molecular Dynamics in a Prototypical Anisotropic Model near the Glass Transition

Dynamics and thermodynamics of molecular systems in the vicinity of the boundary between thermodynamically nonequilibrium glassy and metastable supercooled liquid states are still incompletely explored and their theoretical and simulation models are imperfect despite many previous efforts. Among them, the role of total system entropy, configurational entropy, and excess entropy in the temperature-pressure or temperature-density dependence of global molecular dynamics (MD) timescale relevant to the glass transition is an essential topic intensively studied for over half of a century. By exploiting a well-known simple ellipsoidal model recently successfully applied to simulate the supercooled liquid state and the glass transition, we gain a new insight into the different views on the relationship between entropy and relaxation dynamics of glass formers, showing the molecular grounds for the entropy scaling of global MD timescale. Our simulations in the anisotropic model of supercooled liquid, which involves only translational and rotational degrees of freedom, give evidence that the total system entropy is sufficient to scale global MD timescale. It complies with the scaling effect on relaxation dynamics exerted by the configurational entropy defined as the total entropy diminished by vibrational contributions, which was earlier discovered for measurement data collected near the glass transition. Moreover, we argue that such a scaling behavior is not possible to achieve by using the excess entropy that is in excess of the ideal gas entropy, which is contrary to the results earlier suggested within the framework of simple isotropic models of supercooled liquids. Thus, our findings also warn against an excessive reliance on isotropic models in theoretical interpretations of molecular phenomena, despite their simplicity and popularity, because they may reflect improperly various physicochemical properties of glass formers.

Hot dense silica glass with ultrahigh elastic moduli

Silicate and oxide glasses are often chemically doped with a variety of cations to tune for desirable properties in technological applications, but their performances are often limited by relatively lower mechanical and elastic properties. Finding a new route to synthesize silica-based glasses with high elastic and mechanical properties needs to be explored. Here, we report a dense SiO₂-glass with ultra-high elastic moduli using sound velocity measurements by Brillouin scattering up to 72 GPa at 300 K. High-temperature measurements were performed up to 63 GPa at 750 K and 59 GPa at 1000 K. Compared to compression at 300 K, elevated temperature helps compressed SiO₂-glass effectively overcome the kinetic barrier to undergo permanent densification with enhanced coordination number and connectivity. This hot compressed SiO₂-glass exhibits a substantially high bulk modulus of 361–429 GPa which is at least 2–3 times greater than the metallic, oxide, and silicate glasses at ambient conditions. Its Poisson’s ratio, an indicator for the packing efficiency, is comparable to the metallic glasses. Even after temperature quench and decompression to ambient conditions, the SiO₂-glass retains some of its unique properties at compression and possesses a Poisson’s ratio of 0.248(11). In addition to chemical alternatives in glass syntheses, coupled compression and heating treatments can be an effective means to enhance mechanical and elastic properties in high-performance glasses.

Beyond the Average: Spatial and Temporal Fluctuations in Oxide Glass-Forming Systems

Atomic structure dictates the performance of all materials systems; the characteristic of disordered materials is the significance of spatial and temporal fluctuations on composition-structure-property-performance relationships. Glass has a disordered atomic arrangement, which induces localized distributions in physical properties that are conventionally defined by average values. Quantifying these statistical distributions (including variances, fluctuations, and heterogeneities) is necessary to describe the complexity of glass-forming systems. Only recently have rigorous theories been developed to predict heterogeneities to manipulate and optimize glass properties. This article provides a comprehensive review of experimental, computational, and theoretical approaches to characterize and demonstrate the effects of short-, medium-, and long-range statistical fluctuations on physical properties (e.g., thermodynamic, kinetic, mechanical, and optical) and processes (e.g., relaxation, crystallization, and phase separation), focusing primarily on commercially relevant oxide glasses. Rigorous investigations of fluctuations enable researchers to improve the fundamental understanding of the chemistry and physics governing glass-forming systems and optimize structure-property-performance relationships for next-generation technological applications of glass, including damage-resistant electronic displays, safer pharmaceutical vials to store and transport vaccines, and lower-attenuation fiber optics. We invite the reader to join us in exploring what can be discovered by going beyond the average.

Dynamics of structural relaxation in bioactive 45S5 glass

Kinetics of physical aging in archetypic 45S5 bioactive silicate glass composition with different types of phase separation are studied in situ below the glass transition temperature (T g). The qualitative nature of aging is found to be almost independent of the structural differences on the micrometer scale. A well-expressed step-like behavior in the enthalpy recovery kinetics is observed for aging temperatures T a ∼ 0.90T g and T a ∼ 0.85T g, which, however, disappears when the aging occurs at T a ∼ 0.95T g. The overall kinetics are described by a stretched-exponential function with stretching exponent close to 3/7 at T a ∼ 0.95T g, and 1/3 when the aging temperature drops to ∼0.90T g and below. The values correlate well with the predictions of Phillips' diffusion-to-traps and percolating fractals models. Appearance of step-like behavior at larger departure from T g is attributed to the hierarchical scheme of approaching equilibrium based on an alignment-shrinkage mechanism of physical aging proposed earlier for chalcogenide glasses.

Reply to "Comment on 'Glass Transition, Crystallization of Glass-Forming Melts, and Entropy"' by Zanotto and Mauro

A response is given to a comment of Zanotto and Mauro on our paper published in Entropy 20, 103 (2018). Our arguments presented in this paper are widely ignored by them, and no new considerations are outlined in the comment, which would require a revision of our conclusions. For this reason, we restrict ourselves here to a brief response, supplementing it by some additional arguments in favor of our point of view not included in our above-cited paper.

Comment on "Glass Transition, Crystallization of Glass-Forming Melts, and Entropy" Entropy 2018, 20, 103

In a recent article, Schmelzer and Tropin [Entropy2018, 20, 103] presented a critique of several aspects of modern glass science, including various features of glass transition and relaxation, crystallization, and the definition of glass itself. We argue that these criticisms are at odds with well-accepted knowledge in the field from both theory and experiments. The objective of this short comment is to clarify several of these issues.

Glass Transition, Crystallization of Glass-Forming Melts, and Entropy

A critical analysis of possible (including some newly proposed) definitions of the vitreous state and the glass transition is performed and an overview of kinetic criteria of vitrification is presented. On the basis of these results, recent controversial discussions on the possible values of the residual entropy of glasses are reviewed. Our conclusion is that the treatment of vitrification as a process of continuously breaking ergodicity with entropy loss and a residual entropy tending to zero in the limit of zero absolute temperature is in disagreement with the absolute majority of experimental and theoretical investigations of this process and the nature of the vitreous state. This conclusion is illustrated by model computations. In addition to the main conclusion derived from these computations, they are employed as a test for several suggestions concerning the behavior of thermodynamic coefficients in the glass transition range. Further, a brief review is given on possible ways of resolving the Kauzmann paradox and its implications with respect to the validity of the third law of thermodynamics. It is shown that neither in its primary formulations nor in its consequences does the Kauzmann paradox result in contradictions with any basic laws of nature. Such contradictions are excluded by either crystallization (not associated with a pseudospinodal as suggested by Kauzmann) or a conventional (and not an ideal) glass transition. Some further so far widely unexplored directions of research on the interplay between crystallization and glass transition are anticipated, in which entropy may play—beyond the topics widely discussed and reviewed here—a major role.

Geometric structure and information change in phase transitions

We propose a toy model for a cyclic order-disorder transition and introduce a geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of forward and backward processes (FPs and BPs) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent probability density functions (PDFs) and the information length L, which is the total number of different states that a system undergoes during the transition. Time-dependent PDFs during transient relaxation exhibit strikingly different behavior in FPs and BPs. In particular, FPs driven by instability undergo the broadening of the PDF with a large increase in fluctuations before the transition to the ordered state accompanied by narrowing the PDF width. During this stage, we identify an interesting geodesic solution accompanied by the self-regulation between the growth and nonlinear damping where the time scale τ of information change is constant in time, independent of the strength of the stochastic noise. In comparison, BPs are mainly driven by the macroscopic motion due to the movement of the PDF peak. The total information length L between initial and final states is much larger in BPs than in FPs, increasing linearly with the deviation γ of a control parameter from the critical state in BPs while increasing logarithmically with γ in FPs. L scales as |lnD| and D^{-1/2} in FPs and BPs, respectively, where D measures the strength of the stochastic forcing. These differing scalings with γ and D suggest a great utility of L in capturing different underlying processes, specifically, diffusion vs advection in phase transition by geometry. We discuss physical origins of these scalings and comment on implications of our results for bistable systems undergoing repeated order-disorder transitions (e.g., fitness).

Thermal response of nonequilibrium RC circuits

We analyze experimental data obtained from an electrical circuit having components at different temperatures, showing how to predict its response to temperature variations. This illustrates in detail how to utilize a recent linear response theory for nonequilibrium overdamped stochastic systems. To validate these results, we introduce a reweighting procedure that mimics the actual realization of the perturbation and allows extracting the susceptibility of the system from steady-state data. This procedure is closely related to other fluctuation-response relations based on the knowledge of the steady-state probability distribution. As an example, we show that the nonequilibrium heat capacity in general does not correspond to the correlation between the energy of the system and the heat flowing into it. Rather, also nondissipative aspects are relevant in the nonequilibrium fluctuation-response relations.

A medium range order structural connection to the configurational heat capacity of borate–silicate mixed glasses

Intermediate range order (IRO) structures have a major impact on the composition dependence of the configurational heat capacity of glass.

A stochastic study of electron transfer kinetics in nano-particulate photocatalysis: a comparison of the quasi-equilibrium approximation with a random walking model

A stochastic study was performed in this research, which showed that electron transport to photocatalytic centers cannot reach a quasi-equilibrium state during photocatalysis.

Thermodynamic properties and entropy scaling law for diffusivity in soft spheres

The purely repulsive soft-sphere system, where the interaction potential is inversely proportional to the pair separation raised to the power n, is considered. The Laplace transform technique is used to derive its thermodynamic properties in terms of the potential energy and its density derivative obtained from molecular dynamics simulations. The derived expressions provide an analytic framework with which to explore soft-sphere thermodynamics across the whole softness-density fluid domain. The trends in the isochoric and isobaric heat capacity, thermal expansion coefficient, isothermal and adiabatic bulk moduli, Grüneisen parameter, isothermal pressure, and the Joule-Thomson coefficient as a function of fluid density and potential softness are described using these formulas supplemented by the simulation-derived equation of state. At low densities a minimum in the isobaric heat capacity with density is found, which is a new feature for a purely repulsive pair interaction. The hard-sphere and n = 3 limits are obtained, and the low density limit specified analytically for any n is discussed. The softness dependence of calculated quantities indicates freezing criteria based on features of the radial distribution function or derived functions of it are not expected to be universal. A new and accurate formula linking the self-diffusion coefficient to the excess entropy for the entire fluid softness-density domain is proposed, which incorporates the kinetic theory solution for the low density limit and an entropy-dependent function in an exponential form. The thermodynamic properties (or their derivatives), structural quantities, and diffusion coefficient indicate that three regions specified by a convex, concave, and intermediate density dependence can be expected as a function of n, with a narrow transition region within the range 5 < n < 8.

Ergodicity, configurational entropy and free energy in pigment solutions and plant photosystems: influence of excited state lifetime

We examine ergodicity and configurational entropy for a dilute pigment solution and for a suspension of plant photosystem particles in which both ground and excited state pigments are present. It is concluded that the pigment solution, due to the extreme brevity of the excited state lifetime, is non-ergodic and the configurational entropy approaches zero. Conversely, due to the rapid energy transfer among pigments, each photosystem is ergodic and the configurational entropy is positive. This decreases the free energy of the single photosystem pigment array by a small amount. On the other hand, the suspension of photosystems is non-ergodic and the configurational entropy approaches zero. The overall configurational entropy which, in principle, includes contributions from both the single excited photosystems and the suspension which contains excited photosystems, also approaches zero. Thus the configurational entropy upon photon absorption by either a pigment solution or a suspension of photosystem particles is approximately zero.

Molecular dynamics study of one-component soft-core system: thermodynamic properties in the supercooled liquid and glassy states

Molecular dynamics simulations were performed to study the thermal properties of a supercooled liquid near the glass transition regime and of glasses in a one-component soft-core system with the pair potential φn(r) = ɛ(σ/r)(n), in which n = 12. The results are examined along a phase diagram, in which the compressibility factor defined by [Formula: see text] is plotted against the reduced density ρ* = ρ(ɛ/kBT)(3/n) (or the reduced temperature T* = ρ*(-n/3)). Similarly, a time-dependent dynamical compressibility factor can be plotted against the time-dependent reduced density [Formula: see text] (or the reduced time-dependent temperature). Analytical expressions of the specific heats CV and CP and of the entropy, S, were obtained as a function of [Formula: see text] or of the scaled potential U*. Even for a rapid cooling process, the CV values are found to be affected by non-equilibrium relaxations in the [Formula: see text] region, where [Formula: see text] is the given initial value of [Formula: see text]. The problem of the Kauzmann paradox is discussed using these expressions. The fluctuation of the time-dependent temperature, Tt*, which determines CV, is characterized by the spectra that are obtained by multitaper methods. The thermal fluctuation along the non-equilibrium relaxation under NVE conditions was also examined.

Topological principles of borosilicate glass chemistry

Borosilicate glasses display a rich complexity of chemical behavior depending on the details of their composition and thermal history. Noted for their high chemical durability and thermal shock resistance, borosilicate glasses have found a variety of important uses from common household and laboratory glassware to high-tech applications such as liquid crystal displays. In this paper, we investigate the topological principles of borosilicate glass chemistry covering the extremes from pure borate to pure silicate end members. Based on NMR measurements, we present a two-state statistical mechanical model of boron speciation in which addition of network modifiers leads to a competition between the formation of nonbridging oxygen and the conversion of boron from trigonal to tetrahedral configuration. Using this model, we derive a detailed topological representation of alkali-alkaline earth-borosilicate glasses that enables the accurate prediction of properties such as glass transition temperature, liquid fragility, and hardness. The modeling approach enables an understanding of the microscopic mechanisms governing macroscopic properties. The implications of the glass topology are discussed in terms of both the temperature and thermal history dependence of the atomic bond constraints and the influence on relaxation behavior. We also observe a nonlinear evolution of the jump in isobaric heat capacity at the glass transition when substituting SiO(2) for B(2)O(3), which can be accurately predicted using a combined topological and thermodynamic modeling approach.

Mechanical relaxation and the notion of time-dependent extent of ergodicity during the glass transition

A postulate that ergodicity and entropy continuously decrease to zero on cooling a liquid to a glassy state was used to support the view that glass has no residual entropy, and the features of mechanical relaxation spectra were cited as proof for the decrease. We investigate whether such spectra and the relaxation isochrones can serve as the proof. We find that an increase in the real component of elastic moduli with an increase in spectral frequency does not indicate continuous loss of ergodicity and entropy, and the spectra do not confirm isothermal glass transition or loss of entropy. Variation in ergodicity and entropy with the spectral frequency has untenable consequences for both thermodynamics and molecular dynamics and implies that, despite a broad distribution of its relaxation times, an equilibrium liquid can be considered as always ergodic. Perturbation from equilibrium used to obtain a spectrum does not have the effect of dynamic freezing and unfreezing, and Maxwell-Voigt models for the mechanical response function have neither the characteristic irreversibility of liquid-glass transition nor are commutable to ergodicity or entropy.

On the reality of the residual entropies of glasses and disordered crystals: counting microstates, calculating fluctuations, and comparing averages

In the course of an on-going debate on whether glasses or disordered crystals should have zero entropy at 0 K, i.e., whether the "residual entropy" assigned to them by calorimetric measurements is real, the view has been expressed by some who hold the zero entropy view that to measure entropy, all or an appreciable number of the microstates that contribute to the entropy must be visited. We show here that the entropy calculated on the basis of the number of microstates visited during any conceivable time of measurement would be underestimated by at least 20 orders of magnitude. We also examine and refute the claim that an ensemble average for glassy systems, which predicts a finite residual entropy, also predicts physically impossible properties. We conclude that calorimetrically measured residual entropies are real.