Liquid alkali metals at the melting point: Structural and dynamical properties

Modified Landau-Placzek ratio of the liquid metal rubidium beyond hydrodynamics

The intensity ratio of the Rayleigh line and the Brillouin lines can be derived within hydrodynamics and is known as the Landau-Placzek (LP) ratio. This ratio is directly related to the ratio of specific heats of the fluid. Within the microscopic wave vector range, which can be probed by inelastic neutron scattering, the intensity ratio for simple liquid metals deviates distinctly from the hydrodynamic prediction of the LP-ratio. We derive the intensity ratio from experimental data of liquid rubidium, which shows an enhanced LP-ratio by a factor 8 compared to the hydrodynamic prediction. This strong deviation indicates a further relaxation process in the microscopic wave vector range beyond hydrodynamics. That relaxation process is the viscoelastic reaction of the simple liquid to density fluctuations. Taking this process into account a modified LP-ratio is able to describe the data quite well.

Sodium ion self-diffusion in molten NaBr probed over different length scales

The single-particle dynamics of sodium ions in molten sodium bromide has been investigated with quasielastic neutron scattering. A detailed and rather extensive data analysis procedure allowed determination of the pure sodium ion dynamics with increasing wave vector. Two different evaluation procedures agree perfectly on the resulting diffusion coefficient of sodium ions on long distances. A simple kinetic theory based on binary collisions of hard spheres is not able to reproduce the sodium diffusion coefficient. The derived reduced linewidth from modeling with a Lorentzian spectral function decreases with increasing wave vector towards the first structure factor maximum. That deviation from the hydrodynamic behavior signals the hindrance of the microscopic diffusion process due to the so-called cage effect when microscopic length scales are probed in a dense fluid. The observed quadratic wave-number-dependent decrease might be evidence for a coupling to density fluctuations as the source of the changes in the diffusion process. The results indicate that in the molten salt NaBr near the melting point the self-diffusion process might be governed by similar processes as already observed in dense metallic liquids.

Wave vector dependent damping of THz collective modes in a liquid metal

Well-defined damped collective modes have been observed in liquid metals over a wide range of wave vectors. Hydrodynamics predicts that viscosity and thermal conductivity are the cause for the damping of the collective modes. Here we present experimental data from neutron spectroscopy on the damping of collective modes of liquid rubidium over a wide range of wave vectors. We propose a phenomenological model derived from generalized hydrodynamics to describe the damping of the modes and the evolution with increasing wave vector based on the viscoelastic picture of liquid response. As necessary ingredients a wave vector dependent high frequency shear modulus and shear relaxation time appear. We obtain a remarkable good agreement on a quantitative basis between experiment and calculation over a wide range of wave vectors. The emergent picture is that the lifetime of the collective modes in the THz regime is mainly limited through the diffusion of momentum. The proposed methodology might be applicable to a wide range of liquids.

A search for manifestation of two types of collective excitations in dynamic structure of a liquid metal: Ab initio study of collective excitations in liquid Na

Using a combination of ab initio molecular dynamics and several fit models for dynamic structure of liquid metals, we explore an issue of possible manifestation of non-acoustic collective excitations in longitudinal dynamics having liquid Na as a case study. A model with two damped harmonic oscillators (DHOs) in time domain is used for analysis of the density-density time correlation functions. Another similar model with two propagating contributions and three lowest exact sum rules is considered, as well as an extended hydrodynamic model known as thermo-viscoelastic one which permits two types of propagating modes outside the hydrodynamic region to be used for comparison with ab initio obtained time correlation functions and calculations of dispersions of collective excitations. Our results do not support recent suggestions that, even in simple liquid metals, non-hydrodynamics transverse excitations contribute to the longitudinal collective dynamics and can be detected as a DHO-like spectral shape at their transverse frequency. We found that the thermo-viscoelastic dynamic model permits perfect description of the density-density and current-current time correlation functions of the liquid Na in a wide range of wave numbers, which implies that the origin of the non-hydrodynamic collective excitations contributing to longitudinal dynamics can be short-wavelength heat waves.

A new approach for estimating the density of liquids

We propose a novel approach with which to estimate the density of liquids. The approach is based on the assumption that the systems would be structurally similar when viewed at around the length scale (inverse wavenumber) of the first peak of the structure factor, unless their thermodynamic states differ significantly. The assumption was implemented via a similarity transformation to the radial distribution function to extract the density from the structure factor of a reference state with a known density. The method was first tested using two model liquids, and could predict the densities within an error of several percent unless the state in question differed significantly from the reference state. The method was then applied to related real liquids, and satisfactory results were obtained for predicted densities. The possibility of applying the method to amorphous materials is discussed.

Freezing and melting line invariants of the Lennard-Jones system

The invariance of several structural and dynamical properties of the Lennard-Jones (LJ) system along the freezing and melting lines is interpreted in terms of isomorph theory. First the freezing/melting lines of the LJ system are shown to be approximated by isomorphs. Then we show that the invariants observed along the freezing and melting isomorphs are also observed on other isomorphs in the liquid and crystalline phases. The structure is probed by the radial distribution function and the structure factor and dynamics are probed by the mean-square displacement, the intermediate scattering function, and the shear viscosity. Studying these properties with reference to isomorph theory explains why the known single-phase melting criteria hold, e.g., the Hansen-Verlet and the Lindemann criteria, and why the Andrade equation for the viscosity at freezing applies, e.g., for most liquid metals. Our conclusion is that these empirical rules and invariants can all be understood from isomorph theory and that the invariants are not peculiar to the freezing and melting lines, but hold along all isomorphs.

Br diffusion in molten NaBr explored by coherent quasielastic neutron scattering

Molten sodium bromide has been investigated by quasielastic neutron scattering focusing on the wave vector range around the first structure factor peak. The linewidth of the scattering function shows a narrowing around the wave number of the structure factor peak, known as deGennes narrowing. In a monatomic system, this narrowing or in the time domain slowing down, has been related to a self-diffusion process of the caged particle. Here we show that this methodology can be applied to the molten alkali halide NaBr. The incoherent scattering from the sodium ions at small wave vectors provides the self-diffusion coefficient of sodium and the dynamics of bromine ions can be studied at wave numbers around the structure factor peak. With input from molecular dynamics simulations on the partial structure factors, diffusion coefficients of the bromine ions can be obtained. These experimentally derived diffusion coefficients are in good agreement with molecular dynamics simulation results. This methodology to extract self-diffusion coefficients from coherent quasielastic neutron scattering is applicable to binary fluids in general when one particle dominates the scattering response at the structure factor maximum.

An effective fitting scheme for the dynamic structure of pure liquids

A scheme of analysis for the dynamic structure functions in pure liquids is presented which can be implemented with both experimental and simulation data. Expressions for contributions of relaxing and propagating modes proposed earlier in the framework of the generalized collective modes approach are optimized in order to strictly fulfil three among the required sum-rules. The method is applied to simulation data for liquid cesium, the description of which appears to only require one relaxing and one propagating mode in the investigated wavevector range. These expressions are able to account for the dynamics in both the hydrodynamic and the kinetic regimes, being quantitatively accurate up to the onset of the first peak of the static structure factor and qualitatively beyond. Features of the modes can thus be obtained easily, without resorting to heavy formalism. The scheme of analysis can be straightforwardly extended to account for a higher number of relaxing and propagating modes.

Statistical mechanics model for the transit free energy of monatomic liquids

In applying vibration-transit (V-T) theory of liquid dynamics to the thermodynamic properties of monatomic liquids, the point has been reached where an improved model is needed for the small (approximately 10%) transit contribution. Toward this goal, an analysis of the available high-temperature experimental entropy data for elemental liquids was recently completed [D. C. Wallace, E. D. Chisolm, and N. Bock, Phys. Rev. E 79, 051201 (2009)]. This analysis yields a common curve of transit entropy vs T/thetatr, where T is temperature and thetatr is a scaling temperature for each element. In the present paper, a statistical mechanics model is constructed for the transit partition function, and is calibrated to the experimental transit entropy curve. The model has two scalar parameters, and captures the temperature scaling of experiment. The calibrated model fits the experimental liquid entropy to high accuracy at all temperatures. With no additional parameters, the model also agrees with both experiment and molecular dynamics for the internal energy vs. T for Na. With the calibrated transit model, V-T theory provides equations subject to ab initio evaluation for thermodynamic properties of monatomic liquids. This will allow the range of applicability of the theory, and its overall accuracy, to be determined. More generally, the hypothesis of V-T theory, which divides the many-atom potential energy valleys into random and symmetric classes, can also be tested for its application beyond monatomic systems.

Vibration-transit theory of self-dynamic response in a monatomic liquid

A theoretical model for self-dynamic response is developed using vibration-transit theory, and is applied to liquid sodium at all wave vectors q from the hydrodynamic regime to the free particle limit. In this theory the zeroth-order Hamiltonian describes the vibrational motion in a single random valley harmonically extended to infinity. This Hamiltonian is tractable, is evaluated a priori for monatomic liquids, and the same Hamiltonian (the same set of eigenvalues and eigenvectors) is used for equilibrium and nonequilibrium theory. Here, for the self-intermediate scattering function F;{s}(q,t) , we find the vibrational contribution is in near perfect agreement with molecular dynamics (MD) through short and intermediate times, at all q . This is direct confirmation that normal mode vibrational correlations are present in the motion of the liquid state. The primary transit effect is the diffusive motion of the vibrational equilibrium positions, as the liquid transits rapidly among random valleys. This motion is modeled as a standard random walk, and the resulting theoretical F;{s}(q,t) is in excellent agreement with MD results at all q and t . In the limit q-->infinity , the theory automatically exhibits the correct approach to the free-particle limit. Also, in the limit q-->0 , the hydrodynamic limit emerges as well. In contrast to the benchmark theories of generalized hydrodynamics and mode coupling, the present theory is near a priori, while achieving modestly better accuracy. Therefore, in our view, it constitutes an improvement over the traditional theories.

Single-random-valley approximation in vibration-transit theory of liquid dynamics

The first goal of vibration-transit theory is to be able to calculate from a tractable partition function and without adjustable parameters the thermodynamic properties of the elemental monatomic liquids. The key hypothesis is that the random class of potential energy valleys dominates the statistical mechanics of the liquid at temperatures above melting T approximately greater than Tm and that these valleys are macroscopically uniform in the thermodynamic limit. This allows us to use a single random valley to calculate the vibrational contribution to liquid properties, exactly in the thermodynamic limit, and as an approximation at finite number of particles N . This approximation is tested here for liquid Na with a physically realistic potential based on electronic structure theory. Steepest descent quenches were made from the molecular dynamics equilibrium liquid (N=500) at temperatures from 0.90Tm to 3.31Tm, and six potential parameters were calculated for each structure, namely, the potential energy and five principal moments of the vibrational frequency distribution. The results show temperature-independent means and small standard deviations for all potential parameters, consistent with random valley uniformity at N-->infinity, and with finite- N broadening at N=500. The expected error in the single random valley approximation for Na at N=500 and T approximately greater than Tm is 0.1% for the entropy and 0.5% for the internal energy, negligible in the current development of liquid dynamics theory. In related quench studies of recent years, the common finding of nearly temperature-independent means of structural potential energy properties at T approximately greater than Tm suggests that the single random valley approximation might also apply to systems more complicated than the elemental liquids.

State dependent particle dynamics in liquid alkali metals

This paper gives a survey of the particle dynamics in the liquid alkali metals observed with inelastic x-ray and neutron scattering experiments. Liquid rubidium and sodium are chosen as model fluids to represent the behaviour of this group of fluids. In the dense metallic monatomic melt the microscopic dynamics is characterized by collective excitations similar to those in the corresponding solids. The collective particle behaviour is appropriately described using a memory function formalism with two relaxation channels for the density correlation. A similar behaviour is found for the single particle motion where again two relaxation mechanisms are needed to accurately reproduce the experimental findings. Special emphasis is given to the density dependence of the particle dynamics. An interesting issue in liquid metals is the metal to non-metal transition, which is observed if the fluid is sufficiently expanded with increasing temperature and pressure. This causes distinct variations in the interparticle interactions, which feed back onto the motional behaviour. The associated variations in structure and dynamics are reflected in the shape of the scattering laws. The experimentally observed features are discussed and compared with simple models and with the results from computer simulations.

Velocity autocorrelation function for simple liquids and its application to liquid metals and alloys

The single-particle Brownian motion in simple monatomic liquids has been described with the help of the equation of motion in terms of velocity autocorrelation function psi(t) , where the particle is assumed to diffuse in a mean-time-dependent field. The equation of motion which is a second order differential equation in psi(t) has been solved in terms of nth order Bessel functions of the first and second kind. The solution is, in fact, the generalization of the solution given by Glass and Rice (GR) [Phys. Rev. 176, 239 (1968)] for low density fluids. The generalized solution is an improvisation over the one by GR and the results demonstrate that, for high density liquids such as liquid metals where the GR solution fails to give a reasonable account of psi(t) , the generalized solution could be very useful to calculate psi(t) . The present approach has been applied to the liquid alkali metals and binary alloys. The results are found to be in close agreement with the molecular dynamics results.

Relaxation time scales in collective dynamics of liquid alkali metals

In this paper the investigation of the dynamical processes of liquid alkali metals is executed by analyzing the time scales of relaxation processes in liquids. The obtained theoretical dynamic structure factor S(k,omega) for the case of liquid lithium is found to be in excellent agreement with the recently received inelastic x-ray scattering data. The comparison and interrelation with other theories are given here. Finally, an important part of this paper is the confirmation of the scale uniformity of the dynamic processes in liquid alkali metals predicted by some previous molecular dynamic simulation studies.

Excess entropy scaling for the diffusion coefficient in expanded liquid metals

Molecular-dynamics simulation is used to compute the pair correlation function and the velocity autocorrelation function of Cs and Rb along the liquid-vapor coexistence curve, from which the excess entropy S(ex) and the diffusion coefficient D are deduced. The numerical results of both physical properties are correlated and a scaling law between the excess entropy and the reduced diffusion coefficient D(*)(=D/D(0)) is investigated for different expressions of the reduction parameter D(0). The choice of thermodynamic states along the liquid--vapor coexistence curve gives us the possibility to extend the investigation of the relation between the reduced diffusion coefficient and the excess entropy over a wide area and to test the adequacy of the scaling law confidently.

Collective dynamics in molten potassium: An inelastic x-ray scattering study

The high-frequency collective dynamics of molten potassium has been investigated by inelastic x-ray scattering, disclosing an energy/momentum transfer region unreachable by previous inelastic neutron scattering (INS) experiments. We find that a two-step relaxation scenario, similar to that found in other liquid metals, applies to liquid potassium. In particular, we show how the sound velocity determined by INS experiments, exceeding the hydrodynamic value by approximately 30%, is the higher limit of a speedup, located in the momentum region 1 < Q < 3 nm(-1), which marks the departure from the isothermal value. We point out how this phenomenology is the consequence of a microscopic relaxation process that, in turn, can be traced back to the presence of "instantaneous" disorder, rather than to the crossover from a liquid to solidlike response.

High-frequency acoustic modes in liquid gallium at the melting point

The microscopic dynamics in liquid gallium at melting has been studied by inelastic x-ray scattering. We demonstrate the existence of acousticlike modes up to wave vectors above one-half of the first maximum of the static structure factor, at variance with earlier results from inelastic neutron scattering [F. J. Bermejo et al., Phys. Rev. E 49, 3133 (1994)]. Despite structural (extremely rich polymorphism) and electronic (mixed valence) peculiarities, the collective dynamics is strikingly similar to the one of van der Waals and metallic fluids. This result speaks in favor of the universality of the short time dynamics in monatomic liquids rather than of system-specific dynamics.

Evidence of short-time dynamical correlations in simple liquids

We report a molecular dynamics study of the collective dynamics of a simple monatomic liquid--interacting through a two-body potential that mimics that of lithium--across the liquid-glass transition. In the glassy phase we find evidences of a fast relaxation process similar to that recently found in Lennard-Jones glasses. The origin of this process is ascribed to the topological disorder, i.e., to the dephasing of the different momentum Q Fourier components of the actual normal modes of vibration of the disordered structure. More important, we find that the fast relaxation persists in the liquid phase with almost no temperature dependence of its characteristic parameters (strength and relaxation time). We conclude, therefore, that in the liquid phase well above the melting point, at variance with the usual assumption of uncorrelated binary collisions, the short time particle motion is strongly correlated and can be described via a normal mode expansion of the atomic dynamics.

Inelastic x-ray scattering study of the collective dynamics in liquid sodium

Inelastic x-ray scattering data have been collected for liquid sodium at T=390 K, i.e., slightly above the melting point. Owing to the very high instrumental resolution, pushed up to 1.5 meV, it has been possible to determine accurately the dynamic structure factor S(Q,omega) in a wide wave-vector range, 1.5-15 nm(-1), and to investigate on the dynamical processes underlying the collective dynamics. A detailed analysis of the line shape of S(Q,omega), similarly to other liquid metals, reveals the coexistence of two different relaxation processes with slow and fast characteristic time scales. The present data lead to the conclusion that (i) the picture of the relaxation mechanism based on a simple viscoelastic model fails and (ii) although the comparison with other liquid metals reveals similar behavior, the data do not exhibit an exact scaling law as the principle of the corresponding state would predict.

Self-dynamic structure factor of dense liquids: theory and simulation

The self-intermediate dynamic structure factor Fs(k,t) of liquid lithium near the melting temperature is calculated by molecular dynamics. The results are compared with the predictions of several theoretical approaches, paying special attention to the Lovesey model and the Wahnström and Sjögren mode-coupling theory. To this end the results for the Fs(k,t) second memory function predicted by both models are compared with the ones calculated from the simulations.

Evidence of two viscous relaxation processes in the collective dynamics of liquid lithium

New inelastic x-ray scattering experiments have been performed on liquid lithium in a wide wave vector range. With respect to the previous measurements, the instrumental resolution, improved up to 1. 5 meV, allows one to accurately investigate the dynamical processes determining the observed shape of the dynamic structure factor S(Q, omega). A detailed analysis of the line shapes shows the coexistence of relaxation processes with both slow and fast characteristic time scales, and therefore shows that pictures of the relaxation mechanisms based on a simple viscoelastic model must be abandoned.

Dynamic properties of Lennard-Jones fluids and liquid metals

The dependence of the dynamic properties of liquid metals and Lennard-Jones fluids on the characteristics of the interaction potentials is analyzed. Molecular-dynamics simulations of liquids in analogous conditions but assuming that their particles interact either through a Lennard-Jones or a liquid-metal potential were carried out. The Lennard-Jones potentials were chosen so that both the effective size of the particles and the depth of the potential well were very close to those of the liquid-metal potentials. In order to investigate the extent to which the dynamic properties of liquids depend on the short-range attractive interactions as well as on the softness of the potential cores, molecular-dynamics simulations of the same systems but assuming purely repulsive interactions with the same potential cores were also performed. The study includes both single-particle dynamic properties, such as the velocity autocorrelation functions, and collective dynamic properties, such as the intermediate scattering functions, the dynamic structure factors, the longitudinal and transverse current correlations, and the transport coefficients.