Conjugate gradient filtering of instantaneous normal modes, saddles on the energy landscape, and diffusion in liquids

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.

What does the instantaneous normal mode spectrum tell us about dynamical heterogeneity in glass-forming fluids?

We examine the instantaneous normal mode spectrum of model metallic and polymeric glass-forming liquids. We focus on the localized modes in the unstable part of the spectrum [unstable localized (UL) modes] and find that the particles making the dominant contribution to the participation ratio form clusters that grow upon cooling in a fashion similar to the dynamical heterogeneity in glass-forming fluids, i.e., highly mobile (or immobile) particles form clusters that grow upon cooling; however, a comparison of the UL mode clusters to the mobile and immobile particle clusters indicates that they are distinct entities. We also show that the cluster size provides an alternate method to distinguish localized and delocalized modes, offering a significant practical advantage over the finite-size scaling approach. We examine the trajectories of particles contributing most to the UL modes and find that they have a slightly enhanced mobility compared to the average, and we determine a characteristic time quantifying the persistence time of this excess mobility. This time scale is proportional to the structural relaxation time τ_α of the fluid, consistent with a prediction by Zwanzig [Phys. Rev. 156, 190 (1967)] for the lifetime of collective excitations in cooled liquids. Evidently, these collective excitations serve to facilitate relaxation but do not actually participate in the motion associated with barrier crossing events governing activated transport. They also serve as a possible concrete realization of the "facilitation" clusters postulated in previous modeling of glass-forming liquids.

Topological properties of the mean-field phi4 model

We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a phi4 mean-field model. We compare the critical energy vc [i.e., the potential energy vT evaluated at the phase transition temperature Tc ] with the energy vtheta at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc >> vtheta, at variance to what has been found in different mean-field and short ranged systems, where the thermodynamic phase transitions take place at vc=vtheta [Phys. Rep. 337, 237 (2000)]]. By direct calculation of the energy vs T of the "inherent saddles," i.e., the saddles visited by the equilibrated system at temperature T , we find that vsTc approximately vtheta. Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather than to a change of the topology of the potential energy surface at T= Tc. Finally, we discuss the approximation involved in our analysis and the generality of our method.

Modeling velocity autocorrelation functions for confined fluids using γ distributions

We propose a model for the short-time dynamics of fluids confined in slit-shaped pores. The model has been developed from the observation that the real lobe of the instantaneous normal mode density of states (INM DOS) closely follows a gamma distribution. By proposing that the density of states of the confined fluid can be represented by a gamma distribution, the resulting velocity autocorrelation function (VACF) is constructed such that it is accurate upto the fourth frequency moment. The proposed model results in an analytical expression for the VACF and relaxation times. The VACFs obtained from the model have been compared with the VACFs obtained from molecular dynamic simulations and INM analysis for fluids confined in slit-shaped pores over a wide range of confinement and temperatures. The model is seen to capture the short-time behavior of the VACF extremely accurately and in this region is superior to the predictions of the VACF obtained from the real lobe of the INM DOS. Although the model predicts a zero self-diffusivity, the predicted relaxation times are in better agreement with the molecular dynamics results when compared with those obtained from the INM theory.

Scaling and universality of inherent structure simulations

In this paper we explore the inherent structures (IS) approach to the dynamics of the East constrained kinetic Ising model. The inherent structures do not capture the nature of the dynamics of many quantities, including the spin autocorrelation function. Simply monitoring the quenched energy fluctuations, i.e., IS energy, results in an oversimplified single order-parameter description of the system's dynamics, but examining other features, such as domain dynamics or normal modes, may give a more complete and useful picture of the dynamics. The universality in the behavior of the IS energy of this model does not reveal nonuniversal features of the kinetics that determine long-time relaxation of the system. As a result, popular functional forms, such as the stretched exponential relaxation or Gaussian distribution of energies, may be a numerical fit to data with little physical justification. Filtering data can be shown to erase features of the system and the resulting quantities resemble more universal functional forms that lack physical insight. These results for the East model have implications for IS simulations of realistic systems and suggest careful analysis including the examination of other potential order parameters is necessary to evaluate the validity of applications of universal and scaling arguments to IS simulations.

Comparison of kinetic Monte Carlo and molecular dynamics simulations of diffusion in a model glass former

We present results from kinetic Monte Carlo (KMC) simulations of diffusion in a model glass former. We find that the diffusion constants obtained from KMC simulations have Arrhenius temperature dependence, while the correct behavior, obtained from molecular dynamics simulations, can be super-Arrhenius. We conclude that the discrepancy is due to undersampling of higher-lying local minima in the KMC runs. We suggest that the relevant connectivity of minima on the potential energy surface is proportional to the energy density of the local minima, which determines the "inherent structure entropy." The changing connectivity with potential energy may produce a correlation between dynamics and thermodynamics.

What does the potential energy landscape tell us about the dynamics of supercooled liquids and glasses?

For a model glass former we demonstrate via computer simulations how macroscopic dynamic quantities can be inferred from a potential energy landscape (PEL) analysis. The essential step is to consider whole superstructures of many PEL minima, called metabasins, rather than single minima. We show that two types of metabasins exist: some allowing for quasifree motion on the PEL (liquidlike), and the others acting as traps (solidlike). The activated, multistep escapes from the latter metabasins are found to dictate the slowing down of dynamics upon cooling over a much broader temperature range than is currently assumed.

Relative pair dynamics in simple supercooled liquids: longitudinal contributions

The pair dynamics of simulated argon samples is investigated at the melting (85 K), supercooled (55 K), and quenched (20 K) liquid states, and in the crystal (20 K) state. Tagged pairs, initially lying in a given shell, were divided into incoming and outgoing groups and followed along simulated trajectories. Over them, specific correlation functions deltaB(r(0);t), involving the pair separation vector projected along its initial value r(0) (longitudinal dynamics), have been evaluated. More or less pronounced oscillations are detected according to the temperature of the thermodynamic states (and, obviously, their solid or liquid nature); for each state, they depend on the initial pair distance r(0), too. The oscillations vanish after few picoseconds (fast dynamics) in the case of crystal, whereas in the supercooled liquid they decay towards a plateau, whose height increases with the temperature. It is shown that the power spectrum of deltaB(r(0);t) practically yields the same density of states (DOS) produced by the pair velocity correlation function. The deltaB(r(0);t) functions obtained from the argon crystal at 20 K produce DOS curves dominated by two main frequency contributions, at about 40 and 60 cm(-1) (Einstein and Debye frequency, respectively). Their shape is quite well reproduced by damped harmonic oscillator-like (DHO) functions vibrating at that frequencies. In liquid states, the deltaB(r(0);t) plateau, that forms after the fast DHO dynamics, accounts for the system diffusivity. The relaxation towards the plateau is modeled by an exponential function whose decay time is comparable with the average vibration period. Evidence that the liquid states conserve a certain memory of the vibrational modes of the crystal is obtained. In these states, the DHO functions at the Einstein and Debye frequencies plus an exponential function cannot reproduce the deltaB(r(0);t) shape. A pronounced shoulder, that forms around 0.5 ps, requires the contribution of a third DHO. In the DOS, it yields a band centered below 20 cm(-1) that produces low frequency DOS excess in comparison with the DOS of the crystal. This contribution is present in liquid and supercooled high temperature states and survives near the temperature of the glass transition whereas the diffusion practically vanishes.

Random energy model for dynamics in supercooled liquids: N dependence

The random energy model (REM) for the critical points (saddles and minima) of the potential energy landscape of liquids is further developed. While thermodynamic properties may be calculated from the unconditional distribution of states G(E), dynamics requires the distribution G(c)(E';E) of energies E' of neighbors connected to a state with energy E. Previously it was shown [T. Keyes, Phys. Rev. E 62, 7905 (2000)] that an uncorrelated REM, G(c)(E';E)=G(E'), is badly behaved in the thermodynamic limit N--> infinity. In the following, a simple expression is obtained for G(c)(E';E), which leads to reasonable N dependences. Results are obtained for the fraction f(u) of imaginary-frequency instantaneous normal modes, the configuration entropy S(c), the distributions of the different-order critical points, and the rate R of escape from a state. Simulation data on f(u)(T) and the density of minima rho(0)(E) in Lennard-Jones and CS2 are fit with the theory, allowing a determination of some model parameters. A universal scaling form for f(u), and a consequent scheme for calculating the mode-coupling temperature T(c) consistently among different materials, is demonstrated. The dependence of the self-diffusion constant D upon R and f(u) is discussed, with the conclusion that D proportional, variant f(u) in deeply supercooled states. The phenomenology of fragile supercooled liquids is interpreted. It is shown that the REM need not have a Kauzmann transition in the relevant temperature range, i.e., above the glass transition.