Thermodynamics and Dynamics for a Model Potential Energy Landscape

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 fresh look at the vibrational and thermodynamic properties of liquids within the soft potential model

Contrary to the case of solids and gases, where Debye theory and kinetic theory offer a good description for most of the physical properties, a complete theoretical understanding of the vibrational and thermodynamic properties of liquids is still missing. Liquids exhibit a vibrational density of states (VDOS) which does not obey Debye law, and a heat capacity which decreases monotonically with temperature, rather than growing as in solids. Despite many attempts, a simple, complete and widely accepted theoretical framework able to formally derive the aforementioned properties has not been found yet. Here, we revisit one of the theoretical proposals, and in particular we re-analyze the properties of liquids within the soft-potential model, originally formulated for glasses. We confirm that, at least at a qualitative level, many characteristic properties of liquids can be rationalized within this model. We discuss the validity of several phenomenological expressions proposed in the literature for the density of unstable modes, and in particular for its temperature and frequency dependence. We discuss the role of negative curvature regions and unstable modes as fundamental ingredients to have a linear in frequency VDOS. Finally, we compute the heat capacity within the soft potential model for liquids and we show that it decreases with temperature, in agreement with experimental and simulation data.

Angell plot from the potential energy landscape perspective

Within the scenario of the potential energy landscape (PEL), a thermodynamic model has been developed to uncover the physics behind the Angell plot. In our model, by separating the barrier distribution in PELs into a Gaussian-like and a power-law form, we obtain a general relationship between the relaxation time and the temperature. The wide range of the experimental data in the Angell plot, as well as the molecular-dynamics data, can be excellently fitted by two characteristic parameters, the effective barrier (ω) and the effective width (σ) of a Gaussian-like distribution. More importantly, the fitted ω and σ^{2} for all glasses are found to have a simple linear relationship within a very narrow band, and fragile and strong glasses are well separated in the ω-σ^{2} plot, which indicates that glassy states appear only in a specific region of the PEL.

Phase space theory of evaporation in neon clusters: the role of quantum effects

Unimolecular evaporation of neon clusters containing between 14 and 148 atoms is theoretically investigated in the framework of phase space theory. Quantum effects are incorporated in the vibrational densities of states, which include both zero-point and anharmonic contributions, and in the possible tunneling through the centrifugal barrier. The evaporation rates, kinetic energy released, and product angular momentum are calculated as a function of excess energy or temperature in the parent cluster and compared to the classical results. Quantum fluctuations are found to generally increase both the kinetic energy released and the angular momentum of the product, but the effects on the rate constants depend nontrivially on the excess energy. These results are interpreted as due to the very few vibrational states available in the product cluster when described quantum mechanically. Because delocalization also leads to much narrower thermal energy distributions, the variations of evaporation observables as a function of canonical temperature appear much less marked than in the microcanonical ensemble. While quantum effects tend to smooth the caloric curve in the product cluster, the melting phase change clearly keeps a signature on these observables. The microcanonical temperature extracted from fitting the kinetic energy released distribution using an improved Arrhenius form further suggests a backbending in the quantum Ne(13) cluster that is absent in the classical system. Finally, in contrast to delocalization effects, quantum tunneling through the centrifugal barrier does not play any appreciable role on the evaporation kinetics of these rather heavy clusters.

Exploring the potential energy landscape of glass-forming systems: from inherent structures via metabasins to macroscopic transport

In this review a systematic analysis of the potential energy landscape (PEL) of glass-forming systems is presented. Starting from the thermodynamics, the route towards the dynamics is elucidated. A key step in this endeavor is the concept of metabasins. The relevant energy scales of the PEL can be characterized. Based on the simulation results for some glass-forming systems one can formulate a relevant model system (ideal Gaussian glass-former) which can be treated analytically. The macroscopic transport can be related to the microscopic hopping processes, using either the strong relation between energy (thermodynamics) and waiting times (dynamics) or, alternatively, the concepts of the continuous-time random walk. The relation to the geometric properties of the PEL is stressed. The emergence of length scales within the PEL approach as well as the nature of finite-size effects is discussed. Furthermore, the PEL view is compared to other approaches describing the glass transition.

Equilibrium density of states and thermodynamic properties of a model glass former

This paper presents an analysis of the thermodynamics of a model glass former. We have performed equilibrium sampling of a popular binary Lennard-Jones model, employing parallel tempering Monte Carlo to cover the crystalline, amorphous, and liquid regions of configuration space. Disconnectivity graphs are used to visualize the potential energy landscape in the vicinity of a crystalline geometry and in an amorphous region of configuration space. The crystalline global minimum is separated from the bulk of the minima by a large potential energy gap, leading to broken ergodicity in conventional simulations. Our sampling reveals crystalline global minima that are lower in potential energy than some of the previous candidates. We present equilibrium thermodynamic properties based on parallel tempering simulations, including heat capacities and free energy profiles, which depend explicitly on the crystal structure. We also report equilibrium melting temperatures.

Inherent Structure Analysis of Protein Folding

An analysis in terms of the inherent structures (IS, local minima) of the multidimensional potential energy landscape is applied to proteins. Detailed calculations are performed for the 46 bead BLN model, which folds into a four-stranded beta-barrel. Enhanced sampling has allowed determination of 239 199 IS states, believed to encompass nearly all the compact, low-energy states, and of well-averaged thermodynamic quantities at low temperature. The density of states shows distinct lobes for compact and extended states, and entropic barriers for the collapse and local ordering transitions. A two-dimensional scatterplot or density of states clearly shows the multifunnel structure of the energy landscape. The anharmonic vibrational free energy is found to play a crucial role in protein folding. The problem of determining the folding transition in a multifunnel system is discussed, and novel indicators of folding are introduced. A particularly clear picture is obtained through the occupation probabilities, pi, of individual low-lying IS, which become finite below the collapse temperature; it is suggested that poor foldability corresponds to a large "misfolding interval" where the excited state pi>0 exceeds that of the native state p0.

Oxygen-deficient perovskites: linking structure, energetics and ion transport

The present review focuses on links between structure, energetics and ion transport in oxygen-deficient perovskite oxides, ABO(3-delta). The perfect long-range order, convenient for interpretations of the structure and properties of ordered materials, is evidently not present in disordered materials and highly defective perovskite oxides are spatially inhomogeneous on an intermediate length scale. Although this makes a fundamental description of these and other disordered materials very difficult, it is becoming increasingly clear that this complexity is often essential for the functional properties. In the present review we advocate a potential energy barrier description of the disordered state in which the possible local (or inherent) structures are seen to correspond to separate local minima on the potential energy surface. We interpret the average structure observed experimentally at any temperature as a time and spatial average of the different local structures which are energetically accessible. The local structure is largely affected by preferences for certain polyhedron coordinations and the oxidation state stability of the transition metals, and the strong long-range electrostatic interactions present in non-stoichiometric oxides imply that only a small fraction of the local energy minima on the potential energy surface are accessible at most temperatures. We will show that models neglecting the spatial inhomogeneity and thus the local structure serve as useful empirical tools for particular purposes, e.g. for understanding the main features of the complex redox properties that are so crucial for many applications of these oxides. The short-range order is on the other hand central for understanding ionic transport. Oxide ion transport involves the transformation of one energetically accessible local structure into another. Thus, strongly correlated transport mechanisms are expected; in addition to the movement of the oxygen ions giving rise to the transport, other ions are involved and even the A and B atoms move appreciably in a cooperative fashion along the transition path. Such strongly correlated or collective ionic migration mechanisms should be considered for fast oxide ion conductors in general and in particular for systems forming superstructures at low temperatures. Structural criteria for fast ion conduction are discussed. A high density of low-lying local energy minima is certainly a prerequisite and for perovskite-related A(2)B(2)O(5) oxides, those containing B atoms that have energetic preference for tetrahedral coordination geometry are especially promising.

Genetic mapping of the distribution of minima on the potential energy surface of disordered systems

We show that genetic algorithms and energy minimizations in combination provide a highly efficient tool for mapping low-energy minima on the erratic and complex potential-energy surfaces of grossly disordered materials. The distribution of energy minima mimics with sufficient accuracy the low-energy portion of the parent distribution of minima and allows accurate calculation of configurational Boltzmann averaged structural and thermodynamic properties in cases where a small fraction of the minima is thermally accessible. The distribution of energy minima obtained using genetic algorithms is biased, and consequently the properties converge slowly at high temperatures. In contrast, an optimized set of a few randomly chosen configurations provides a statistical representable selection for the accurate calculation of configurational-averaged properties at high temperatures, but gives a poor description of the low-energy portion of minima. Thus the properties calculated using the random algorithm are hampered by the presence of systematic errors in cases where a small fraction of the minima is thermally accessible. The inherently slow convergence of both the genetic algorithm and the random selection at intermediate temperatures is tackled by combining the lower fraction of the distribution of minima obtained using genetic algorithms with the intermediate and upper fraction from the random (nonbiased) selection of configurations. For this purpose we introduce a cut-and-scale-type scheme. The resulting combined distribution allows accurate calculation of properties at all temperatures.

Potential-energy landscape of a supercooled liquid and its resemblance to a collection of traps

It is analyzed whether the potential energy landscape of a glass-forming system can be effectively mapped on a random model which is described in statistical terms. For this purpose we generalize the simple trap model of Monthus and Bouchaud [J. Phys. A 29, 3847 (1996)] by dividing the total system into M weakly interacting identical subsystems, each being described in terms of a trap model. The distribution of traps in this extended trap model (ETM) is fully determined by the thermodynamics of the glass former. The dynamics is described by two adjustable parameters, one characterizing the common energy level of the barriers, the other the strength of the interaction. The comparison is performed for the standard binary mixture Lennard-Jones system with 65 particles. The metabasins, identified in our previous work, are chosen as traps. Comparing molecular dynamics simulations of the Lennard-Jones system with Monte Carlo calculations of the ETM allows one to determine the adjustable parameters. Analysis of the first moment of the waiting distribution yields an optimum agreement when choosing M approximately 3 subsystems. Comparison with the second moment of the waiting time distribution, reflecting dynamic heterogeneities, indicates that the sizes of the subsystems may fluctuate.

Two-Gaussian excitations model for the glass transition

We develop a modified "two-state" model with Gaussian widths for the site energies of both ground and excited states, consistent with expectations for a disordered system. The thermodynamic properties of the system are analyzed in configuration space and found to bridge the gap between simple two-state models ("logarithmic" model in configuration space) and the random energy model ("Gaussian" model in configuration space). The Kauzmann singularity given by the random energy model remains for very fragile liquids but is suppressed or eliminated for stronger liquids. The sharp form of constant-volume heat capacity found by recent simulations for binary mixed Lennard-Jones and soft-sphere systems is reproduced by the model, as is the excess entropy and heat capacity of a variety of laboratory systems, strong and fragile. The ideal glass in all cases has a narrow Gaussian, almost invariant among molecular and atomic glassformers, while the excited-state Gaussian depends on the system and its width plays a role in the thermodynamic fragility. The model predicts the possibility of first-order phase transitions for fragile liquids. The analysis of laboratory data for toluene and o-terphenyl indicates that fragile liquids resolve the Kauzmann paradox by a first-order transition from supercooled liquid to ideal-glass state at a temperature between T(g) and Kauzmann temperature extrapolated from experimental data. We stress the importance of the temperature dependence of the energy landscape, predicted by the fluctuation-dissipation theorem, in analyzing the liquid thermodynamics.