Glass polymorphism in TIP4P/2005 water: A description based on the potential energy landscape formalism

Nuclear quantum effects on glassy water under pressure: Vitrification and pressure-induced transformations

We perform classical molecular dynamics (MD) and path-integral MD (PIMD) simulations of H2O and D2O using the q-TIP4P/F model over a wide range of temperatures and pressures to study the nuclear quantum effects (NQEs) on (i) the vitrification of liquid water upon isobaric cooling at different pressures and (ii) pressure-induced transformations at constant temperature between low-density amorphous and high-density amorphous ice (LDA and HDA) and hexagonal ice Ih and HDA. Upon isobaric cooling, classical and quantum H2O and D2O vitrify into a continuum of intermediate amorphous ices (IA), with densities in-between those of LDA and HDA (depending on pressure). Importantly, the density of the IA varies considerably if NQEs are included (similar conclusions hold for ice Ih at all pressures studied). While the structure of the IA is not very sensitive to NQE, the geometry of the hydrogen-bond (HB) is. NQE leads to longer and less linear HB in LDA, HDA, and ice Ih than found in the classical case. Interestingly, the delocalization of the H/D atoms is non-negligible and identical in LDA, HDA, and ice Ih at all pressures studied. Our isothermal compression/decompression MD/PIMD simulations show that classical and quantum H2O and D2O all exhibit LDA-HDA and ice Ih-HDA transformations, consistent with experiments. The inclusion of NQE leads to a softer HB-network, which lowers slightly the LDA/ice Ih-to-HDA transformation pressures. Interestingly, the HB in HDA is longer and less linear than in LDA, which is counterintuitive given that HDA is ≈25% denser than LDA. Overall, our results show that, while classical computer simulations provide the correct qualitative phenomenology of ice and glassy water, NQEs are necessary for a quantitative description.

Supercritical density fluctuations and structural heterogeneity in supercooled water-glycerol microdroplets

Recent experiments and theoretical studies strongly indicate that water exhibits a liquid-liquid phase transition (LLPT) in the supercooled domain. An open question is how the LLPT of water can affect the properties of aqueous solutions. Here, we study the structural and thermodynamic properties of supercooled glycerol-water microdroplets at dilute conditions (χg = 3.2% glycerol mole fraction). The combination of rapid evaporative cooling with femtosecond X-ray scattering allows us to outrun crystallization and gain access to the deeply supercooled regime down to T = 229.3 K. We find that the density fluctuations of the glycerol-water solution or, equivalently, its isothermal compressibility, κT, increases upon cooling. This is confirmed by molecular dynamics simulations, which indicate that the presence of glycerol shifts the temperature of maximum κT from T = 230 K in pure water down to T = 223 K in the solution. Our findings elucidate the interplay between the complex behavior of water, including its LLPT, and the properties of aqueous solutions at low temperatures, which can have practical consequences in cryogenic biological applications and cryopreservation techniques.

Potential energy landscape formalism for quantum molecular liquids

The potential energy landscape (PEL) formalism is a powerful tool within statistical mechanics to study the thermodynamic properties of classical low-temperature liquids and glasses. Recently, the PEL formalism has been extended to liquids/glasses that obey quantum mechanics, but applications have been limited to atomistic model liquids. In this work, we extend the PEL formalism to liquid/glassy water using path-integral molecular dynamics (PIMD) simulations, where nuclear quantum effects (NQE) are included. Our PIMD simulations, based on the q-TIP4P/F water model, show that the PEL of quantum water is both Gaussian and anharmonic. Importantly, the ring-polymers associated to the O/H atoms in the PIMD simulations, collapse at the local minima of the PEL (inherent structures, IS) for both liquid and glassy states. This allows us to calculate, analytically, the IS vibrational density of states (IS-VDOS) of the ring-polymer system using the IS-VDOS of classical water (obtained from classical MD simulations). The role of NQE on the structural properties of liquid/glassy water at various pressures are discussed in detail. Overall, our results demonstrate that the PEL formalism can effectively describe the behavior of molecular liquids at low temperatures and in the glass states, regardless of whether the liquid/glass obeys classical or quantum mechanics.

Potential energy landscape of a flexible water model: Equation of state, configurational entropy, and Adam-Gibbs relationship

The potential energy landscape (PEL) formalism is a tool within statistical mechanics that has been used in the past to calculate the equation of states (EOS) of classical rigid model liquids at low temperatures, where computer simulations may be challenging. In this work, we use classical molecular dynamics (MD) simulations and the PEL formalism to calculate the EOS of the flexible q-TIP4P/F water model. This model exhibits a liquid-liquid critical point (LLCP) in the supercooled regime, at (Pc = 150 MPa, Tc = 190 K, and ρc = 1.04 g/cm3) [using the reaction field technique]. The PEL-EOS of q-TIP4P/F water and the corresponding location of the LLCP are in very good agreement with the MD simulations. We show that the PEL of q-TIP4P/F water is Gaussian, which allows us to calculate the configurational entropy of the system, Sconf. The Sconf of q-TIP4P/F water is surprisingly similar to that reported previously for rigid water models, suggesting that intramolecular flexibility does not necessarily add roughness to the PEL. We also show that the Adam-Gibbs relation, which relates the diffusion coefficient D with Sconf, holds for the flexible q-TIP4P/F water model. Overall, our results indicate that the PEL formalism can be used to study molecular systems that include molecular flexibility, the common case in standard force fields. This is not trivial since the introduction of large bending/stretching mode frequencies is problematic in classical statistical mechanics. For example, as shown previously, we find that such high frequencies lead to unphysical (negative) entropy for q-TIP4P/F water when using classical statistical mechanics (yet, the PEL formalism can be applied successfully).

A continuum of amorphous ices between low-density and high-density amorphous ice

Amorphous ices are usually classified as belonging to low-density or high-density amorphous ice (LDA and HDA) with densities ρLDA ≈ 0.94 g/cm3 and ρHDA ≈ 1.15-1.17 g/cm3. However, a recent experiment crushing hexagonal ice (ball-milling) produced a medium-density amorphous ice (MDA, ρMDA ≈ 1.06 g/cm3) adding complexity to our understanding of amorphous ice and the phase diagram of supercooled water. Motivated by the discovery of MDA, we perform computer simulations where amorphous ices are produced by isobaric cooling and isothermal compression/decompression. Our results show that, depending on the pressure employed, isobaric cooling can generate a continuum of amorphous ices with densities that expand in between those of LDA and HDA (briefly, intermediate amorphous ices, IA). In particular, the IA generated at P ≈ 125 MPa has a remarkably similar density and average structure as MDA, implying that MDA is not unique. Using the potential energy landscape formalism, we provide an intuitive qualitative understanding of the nature of LDA, HDA, and the IA generated at different pressures. In this view, LDA and HDA occupy specific and well-separated regions of the PEL; the IA prepared at P = 125 MPa is located in the intermediate region of the PEL that separates LDA and HDA.

The Importance of Nuclear Quantum Effects on the Thermodynamic and Structural Properties of Low-Density Amorphous Ice: A Comparison with Hexagonal Ice

We study the nuclear quantum effects (NQE) on the thermodynamic properties of low-density amorphous ice (LDA) and hexagonal ice (I_h) at P = 0.1 MPa and T ≥ 25 K. Our results are based on path-integral molecular dynamics (PIMD) and classical MD simulations of H₂O and D₂O using the q-TIP4P/F water model. We show that the inclusion of NQE is necessary to reproduce the experimental properties of LDA and ice I_h. While MD simulations (no NQE) predict that the density ρ(T) of LDA and ice I_h increases monotonically upon cooling, PIMD simulations indicate the presence of a density maximum in LDA and ice I_h. MD and PIMD simulations also predict a qualitatively different T-dependence for the thermal expansion coefficient α_P(T) and bulk modulus B(T) of both LDA and ice I_h. Remarkably, the ρ(T), α_P(T), and B(T) of LDA are practically identical to those of ice I_h. The origin of the observed NQE is due to the delocalization of the H atoms, which is identical in LDA and ice I_h. H atoms delocalize considerably (over a distance ≈ 20-25% of the OH covalent-bond length) and anisotropically (preferentially perpendicular to the OH covalent bond), leading to less linear hydrogen bonds HB (larger HOO angles and longer OO separations) than observed in classical MD simulations.

Anomalous properties in the potential energy landscape of a monatomic liquid across the liquid-gas and liquid-liquid phase transitions

As a liquid approaches the gas state, the properties of the potential energy landscape (PEL) sampled by the system become anomalous. Specifically, (i) the mechanically stable local minima of the PEL [inherent structures (IS)] can exhibit cavitation above the so-called Sastry volume, vS, before the liquid enters the gas phase. In addition, (ii) the pressure of the liquid at the sampled IS [i.e., the PEL equation of state, PIS(v)] develops a spinodal-like minimum at vS. We perform molecular dynamics simulations of a monatomic water-like liquid and verify that points (i) and (ii) hold at high temperatures. However, at low temperatures, cavitation in the liquid and the corresponding IS occurs simultaneously and a Sastry volume cannot be defined. Remarkably, at intermediate/high temperatures, the IS of the liquid can exhibit crystallization, i.e., the liquid regularly visits the regions of the PEL that belong to the crystal state. The model liquid considered also exhibits a liquid-liquid phase transition (LLPT) between a low-density and a high-density liquid (LDL and HDL). By studying the behavior of PIS(v) during the LLPT, we identify a Sastry volume for both LDL and HDL. The HDL Sastry volume marks the onset above which IS are heterogeneous (composed of LDL and HDL particles), analogous to points (i) and (ii) above. However, the relationship between the LDL Sastry volume and the onset of heterogeneous IS is less evident. We conclude by presenting a thermodynamic argument that can explain the behavior of the PEL equation of state PIS(v) across both the liquid-gas phase transition and LLPT.

Nuclear quantum effects on the dynamics and glass behavior of a monatomic liquid with two liquid states

We perform path integral molecular dynamics (PIMD) simulations of a monatomic liquid that exhibits a liquid-liquid phase transition and liquid-liquid critical point. PIMD simulations are performed using different values of Planck's constant h, allowing us to study the behavior of the liquid as nuclear quantum effects (NQE, i.e., atoms delocalization) are introduced, from the classical liquid (h = 0) to increasingly quantum liquids (h > 0). By combining the PIMD simulations with the ring-polymer molecular dynamics method, we also explore the dynamics of the classical and quantum liquids. We find that (i) the glass transition temperature of the low-density liquid (LDL) is anomalous, i.e., Tg LDL(P) decreases upon compression. Instead, (ii) the glass transition temperature of the high-density liquid (HDL) is normal, i.e., Tg HDL(P) increases upon compression. (iii) NQE shift both Tg LDL(P) and Tg HDL(P) toward lower temperatures, but NQE are more pronounced on HDL. We also study the glass behavior of the ring-polymer systems associated with the quantum liquids studied (via the path-integral formulation of statistical mechanics). There are two glass states in all the systems studied, low-density amorphous ice (LDA) and high-density amorphous ice (HDA), which are the glass counterparts of LDL and HDL. In all cases, the pressure-induced LDA-HDA transformation is sharp, reminiscent of a first-order phase transition. In the low-quantum regime, the LDA-HDA transformation is reversible, with identical LDA forms before compression and after decompression. However, in the high-quantum regime, the atoms become more delocalized in the final LDA than in the initial LDA, raising questions on the reversibility of the LDA-HDA transformation.

Evidence of a liquid-liquid phase transition in H[Formula: see text]O and D[Formula: see text]O from path-integral molecular dynamics simulations

We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H[Formula: see text]O and D[Formula: see text]O using the q-TIP4P/F water model over a wide range of temperatures and pressures. The density [Formula: see text], isothermal compressibility [Formula: see text], and self-diffusion coefficients D(T) of H[Formula: see text]O and D[Formula: see text]O are in excellent agreement with available experimental data; the isobaric heat capacity [Formula: see text] obtained from PIMD and MD simulations agree qualitatively well with the experiments. Some of these thermodynamic properties exhibit anomalous maxima upon isobaric cooling, consistent with recent experiments and with the possibility that H[Formula: see text]O and D[Formula: see text]O exhibit a liquid-liquid critical point (LLCP) at low temperatures and positive pressures. The data from PIMD/MD for H[Formula: see text]O and D[Formula: see text]O can be fitted remarkably well using the Two-State-Equation-of-State (TSEOS). Using the TSEOS, we estimate that the LLCP for q-TIP4P/F H[Formula: see text]O, from PIMD simulations, is located at [Formula: see text] MPa, [Formula: see text] K, and [Formula: see text] g/cm[Formula: see text]. Isotope substitution effects are important; the LLCP location in q-TIP4P/F D[Formula: see text]O is estimated to be [Formula: see text] MPa, [Formula: see text] K, and [Formula: see text] g/cm[Formula: see text]. Interestingly, for the water model studied, differences in the LLCP location from PIMD and MD simulations suggest that nuclear quantum effects (i.e., atoms delocalization) play an important role in the thermodynamics of water around the LLCP (from the MD simulations of q-TIP4P/F water, [Formula: see text] MPa, [Formula: see text] K, and [Formula: see text] g/cm[Formula: see text]). Overall, our results strongly support the LLPT scenario to explain water anomalous behavior, independently of the fundamental differences between classical MD and PIMD techniques. The reported values of [Formula: see text] for D[Formula: see text]O and, particularly, H[Formula: see text]O suggest that improved water models are needed for the study of supercooled water.

Advances in the study of supercooled water

In this review, we report recent progress in the field of supercooled water. Due to its uniqueness, water presents numerous anomalies with respect to most simple liquids, showing polyamorphism both in the liquid and in the glassy state. We first describe the thermodynamic scenarios hypothesized for the supercooled region and in particular among them the liquid-liquid critical point scenario that has so far received more experimental evidence. We then review the most recent structural indicators, the two-state model picture of water, and the importance of cooperative effects related to the fact that water is a hydrogen-bonded network liquid. We show throughout the review that water's peculiar properties come into play also when water is in solution, confined, and close to biological molecules. Concerning dynamics, upon mild supercooling water behaves as a fragile glass former following the mode coupling theory, and it turns into a strong glass former upon further cooling. Connections between the slow dynamics and the thermodynamics are discussed. The translational relaxation times of density fluctuations show in fact the fragile-to-strong crossover connected to the thermodynamics arising from the existence of two liquids. When considering also rotations, additional crossovers come to play. Mobility-viscosity decoupling is also discussed in supercooled water and aqueous solutions. Finally, the polyamorphism of glassy water is considered through experimental and simulation results both in bulk and in salty aqueous solutions. Grains and grain boundaries are also discussed.

Structure of High-Pressure Supercooled and Glassy Water

We numerically investigate the structure of deep supercooled and glassy water under pressure, covering the range of densities corresponding to the experimentally produced high- and very-high-density amorphous phases. At T=188  K, a continuous increase in density is observed on varying pressure from 2.5 to 13 kbar, with no signs of first-order transitions. Exploiting a recently proposed approach to the analysis of the radial distribution function-based on topological properties of the hydrogen-bond network-we are able to identify well-defined local geometries that involve pairs of molecules separated by multiple hydrogen bonds, specific to the high- and very-high-density structures.

The role of high-density and low-density amorphous ice on biomolecules at cryogenic temperatures: a case study with polyalanine

Experimental techniques, such as cryo-electron microscopy, require biological samples to be recovered at cryogenic temperatures (T ≈ 100 K) with water being in an amorphous ice state. However, (bulk) water can exist in two amorphous ices at P < 1 GPa, low-density amorphous (LDA) ice at low pressures and high-density amorphous ice (HDA) at high pressures; HDA is ≈20-25% denser than LDA. While fast/plunge cooling at 1 bar brings the sample into LDA, high-pressure cooling (HPC), at sufficiently high pressure, produces HDA. HDA can also be produced by isothermal compression of LDA at cryogenic temperatures. Here, we perform classical molecular dynamics simulations to study the effects of LDA, HDA, and the LDA-HDA transformation on the structure and hydration of a small peptide, polyalanine. We follow thermodynamic paths corresponding to (i) fast/plunge cooling at 1 bar, (ii) HPC at P = 400 MPa, and (iii) compression/decompression cycles at T = 80 K. While process (i) produced LDA in the system, path (iii) produces HDA. Interestingly, the amorphous ice produced in process (ii) is an intermediate amorphous ice (IA) with properties that fall in-between those of LDA and HDA. Remarkably, the structural changes in polyalanine are negligible at all conditions studied (0-2000 MPa, 80-300 K) even when water changes among the low and high-density liquid states as well as the amorphous solids LDA, IA, and HDA. The similarities and differences in the hydration of polyalanine vitrified in LDA, IA, and HDA are described. Since the studied thermodynamic paths are suitable for the cryopreservation of biomolecules, we also study the structure and hydration of polyalanine along isobaric and isochoric heating paths, which can be followed experimentally for the recovery of cryopreserved samples. Upon heating, the structure of polyalanine remains practically unchanged. We conclude with a brief discussion of the practical advantages of (a) using HDA and IA as a cryoprotectant environment (as opposed to LDA), and (b) the use of isochoric heating as a recovery process (as opposed to isobaric heating).

Experimental evidence for glass polymorphism in vitrified water droplets

The question of whether a first-order liquid-to-liquid transition is at the origin of water’s anomalous properties has been controversial since the pioneering experiments by Mishima et al. in 1985 and molecular simulations by Poole et al. in 1992. Since then, experiments aimed at shedding light on this question have been performed using amorphous ices made from crystalline ice, fueling criticism about their crystal-like nature. In the present study, we avoid crystalline ice at any time of the experiment yet still observe a first-order glass-to-glass transition in vitrified liquid droplets. This makes the strong case for glass polymorphism and the direct thermodynamic connection to the liquid-to-liquid transition at higher temperatures, dismissing the criticism voiced for three decades.

The nature of amorphous ices has been debated for more than 35 years. In essence, the question is whether they are related to ice polymorphs or to liquids. The fact that amorphous ices are traditionally prepared from crystalline ice via pressure-induced amorphization has made a clear distinction tricky. In this work, we vitrify liquid droplets through cooling at ≥10⁶ K ⋅ s⁻¹ and pressurize the glassy deposit. We observe a first order–like densification upon pressurization and recover a high-density glass. The two glasses resemble low- and high-density amorphous ice in terms of both structure and thermal properties. Vitrified water shows all features that have been reported for amorphous ices made from crystalline ice. The only difference is that the hyperquenched and pressurized deposit shows slightly different crystallization kinetics to ice I upon heating at ambient pressure. This implies a thermodynamically continuous connection of amorphous ices with liquids, not crystals.

Manifestations of metastable criticality in the long-range structure of model water glasses

Much attention has been devoted to water’s metastable phase behavior, including polyamorphism (multiple amorphous solid phases), and the hypothesized liquid-liquid transition and associated critical point. However, the possible relationship between these phenomena remains incompletely understood. Using molecular dynamics simulations of the realistic TIP4P/2005 model, we found a striking signature of the liquid-liquid critical point in the structure of water glasses, manifested as a pronounced increase in long-range density fluctuations at pressures proximate to the critical pressure. By contrast, these signatures were absent in glasses of two model systems that lack a critical point. We also characterized the departure from equilibrium upon vitrification via the non-equilibrium index; water-like systems exhibited a strong pressure dependence in this metric, whereas simple liquids did not. These results reflect a surprising relationship between the metastable equilibrium phenomenon of liquid-liquid criticality and the non-equilibrium structure of glassy water, with implications for our understanding of water phase behavior and glass physics. Our calculations suggest a possible experimental route to probing the existence of the liquid-liquid transition in water and other fluids.

The subtle connections between water’s supercooled liquid and glassy states are difficult to characterize. Gartner et al. suggest with MD simulations that the long-range structure of glassy water may reflect signatures of water’s debated second critical point in the supercooled liquid.

Glass polyamorphism in gallium: Two amorphous solid states and their transformation on the potential energy landscape

Using the potential energy landscape (PEL) formalism and molecular dynamics simulations, we investigate a phase transformation between two amorphous solid states of gallium, namely, a low-density amorphous solid (LDA) and a high-density amorphous solid (HDA), and compare with its equilibrium counterpart, the liquid-liquid phase transition (LLPT). It is found that on the PEL, the signatures of the out-of-equilibrium LDA-HDA transition are reminiscent of those of the equilibrium LLPT in terms of pressure, inherent structure pressure, inherent structure energy, and shape function, indicating that the LDA-HDA transformation is a first-order-like transition. However, differences are also found between the out-of-equilibrium phase transition and the equilibrium one, for example, the path from LDA to HDA on the PEL cannot be accessed by the path from LDL to HDL. Our results also suggest that the signatures of the out-of-equilibrium transition in gallium are rather general features of systems with an accessible LLPT-not only systems with pairwise interactions but also those with many-body interactions. This finding is of crucial importance for obtaining a deeper understanding of the nature of transitions in the polyamorphic family.

Nuclear quantum effects on the thermodynamic, structural, and dynamical properties of water

We perform path-integral molecular dynamics (PIMD) simulations of H2O and D2O using the q-TIP4P/F model. Simulations are performed at P = 1 bar and over a wide range of temperatures that include the equilibrium (T≥ 273 K) and supercooled (210 ≤T < 273 K) liquid states of water. The densities of both H2O and D2O calculated from PIMD simulations are in excellent agreement with experiments in the equilibrium and supercooled regimes. We also evaluate important thermodynamic response functions, specifically, the thermal expansion coefficient αP(T), isothermal compressibility κT(T), isobaric heat capacity CP(T), and static dielectric constant ε(T). While these properties are in excellent [αP(T) and κT(T)] or semi-quantitative agreement [CP(T) and ε(T)] with experiments in the equilibrium regime, they are increasingly underestimated upon further cooling. It follows that the inclusion of nuclear quantum effects in PIMD simulations of (q-TIP4P/F) water is not sufficient to reproduce the anomalous large fluctuations in density, entropy, and electric dipole moment characteristic of supercooled water. It has been hypothesized that water may exhibit a liquid-liquid critical point (LLCP) in the supercooled regime at P > 1 bar and that such a LLCP generates a maximum in CP(T) and κT(T) at 1 bar. Consistent with this hypothesis and in particular, with experiments, we find a maximum in the κT(T) of q-TIP4P/F light and heavy water at T≈ 230-235 K. No maximum in CP(T) could be detected down to T≥ 210 K. We also calculate the diffusion coefficient D(T) of H2O and D2O using the ring-polymer molecular dynamics (RPMD) technique and find that computer simulations are in remarkable good agreement with experiments at all temperatures studied. The results from RPMD/PIMD simulations are also compared with the corresponding results obtained from classical MD simulations of q-TIP4P/F water where atoms are represented by single interacting sites. Surprisingly, we find minor differences in most of the properties studied, with CP(T), D(T), and structural properties being the only (expected) exceptions.

Experimental observation of the liquid-liquid transition in bulk supercooled water under pressure

We prepared bulk samples of supercooled liquid water under pressure by isochoric heating of high-density amorphous ice to temperatures of 205 ± 10 kelvin, using an infrared femtosecond laser. Because the sample density is preserved during the ultrafast heating, we could estimate an initial internal pressure of 2.5 to 3.5 kilobar in the high-density liquid phase. After heating, the sample expanded rapidly, and we captured the resulting decompression process with femtosecond x-ray laser pulses at different pump-probe delay times. A discontinuous structural change occurred in which low-density liquid domains appeared and grew on time scales between 20 nanoseconds to 3 microseconds, whereas crystallization occurs on time scales of 3 to 50 microseconds. The dynamics of the two processes being separated by more than one order of magnitude provides support for a liquid-liquid transition in bulk supercooled water.

Glass polymorphism and liquid-liquid phase transition in aqueous solutions: experiments and computer simulations

One of the most intriguing anomalies of water is its ability to exist as distinct amorphous ice forms (glass polymorphism or polyamorphism). This resonates well with the possible first-order liquid-liquid phase transition (LLPT) in the supercooled state, where ice is the stable phase. In this Perspective, we review experiments and computer simulations that search for LLPT and polyamorphism in aqueous solutions containing salts and alcohols. Most studies on ionic solutes are devoted to NaCl and LiCl; studies on alcohols have mainly focused on glycerol. Less attention has been paid to protein solutions and hydrophobic solutes, even though they reveal promising avenues. While all solutions show polyamorphism and an LLPT only in dilute, sub-eutectic mixtures, there are differences regarding the nature of the transition. Isocompositional transitions for varying mole fractions are observed in alcohol but not in ionic solutions. This is because water can surround alcohol molecules either in a low- or high-density configuration whereas for ionic solutes, the water ion hydration shell is forced into high-density structures. Consequently, the polyamorphic transition and the LLPT are prevented near the ions, but take place in patches of water within the solutions. We highlight discrepancies and different interpretations within the experimental community as well as the key challenges that need consideration when comparing experiments and simulations. We point out where reinterpretation of past studies helps to draw a unified, consistent picture. In addition to the literature review, we provide original experimental results. A list of eleven open questions that need further consideration is identified.

Distinguishing between Clausius, Boltzmann and Pauling Entropies of Frozen Non-Equilibrium States

In conventional textbook thermodynamics, entropy is a quantity that may be calculated by different methods, for example experimentally from heat capacities (following Clausius) or statistically from numbers of microscopic quantum states (following Boltzmann and Planck). It had turned out that these methods do not necessarily provide mutually consistent results, and for equilibrium systems their difference was explained by introducing a residual zero-point entropy (following Pauling), apparently violating the Nernst theorem. At finite temperatures, associated statistical entropies which count microstates that do not contribute to a body’s heat capacity, differ systematically from Clausius entropy, and are of particular relevance as measures for metastable, frozen-in non-equilibrium structures and for symbolic information processing (following Shannon). In this paper, it is suggested to consider Clausius, Boltzmann, Pauling and Shannon entropies as distinct, though related, physical quantities with different key properties, in order to avoid confusion by loosely speaking about just “entropy” while actually referring to different kinds of it. For instance, zero-point entropy exclusively belongs to Boltzmann rather than Clausius entropy, while the Nernst theorem holds rigorously for Clausius rather than Boltzmann entropy. The discussion of those terms is underpinned by a brief historical review of the emergence of corresponding fundamental thermodynamic concepts.