Eric D. Chisolm

Test of a theoretical equation of state for elemental solids and liquids

We propose a means for constructing highly accurate equations of state (EOS) for elemental solids and liquids essentially from first principles, based upon a particular decomposition of the underlying condensed matter Hamiltonian for the nuclei and electrons. We also point out that at low pressures the neglect of anharmonic and electron-phonon terms, both contained in this formalism, results in errors of less than 5% in the thermal parts of the thermodynamic functions. Then we explicitly display the forms of the remaining terms in the EOS, commenting on the use of experiment and electronic structure theory to evaluate them. We also construct an EOS for aluminum and compare the resulting Hugoniot curve with data up to 5 Mbar, both to illustrate our method and to see whether the approximation of neglecting anharmonicity etc. remains viable to such high pressures. We find a level of agreement with experiment that is consistent with the low-pressure results.

Generalizing the Heisenberg uncertainty relation

The proof of the Heisenberg uncertainty relation is modified to produce two improvements: (a) The resulting inequality is stronger because it includes the covariance between the two observables, and (b) the proof lifts certain restrictions on the state to which the relation is applied, increasing its generality. The restrictions necessary for the standard inequality to apply are not widely known, and they are discussed in detail. The classical analog of the Heisenberg relation is also derived, and the two are compared. Finally, the modified relation is used to address the apparent paradox that eigenfunctions of the z component of angular momentum Lz do not satisfy the φ–Lz Heisenberg relation; the resolution is that the restrictions mentioned above make the usual inequality inapplicable to these states. The modified relation does apply, however, and it is shown to be consistent with explicit calculations.

A Comparison of Theory and Experiment of the Bulk Sound Velocity in Aluminum Using a Two‐Phase EOS

We compute the bulk sound speed along the Hugoniot using a new solid‐liquid two‐phase equation of state (EOS) for aluminum [Chisolm, Crockett, and Wallace, to appear in Phys. Rev. B] and compare with experimental sound speeds from various sources. The experiment extends from the crystal through the entire solid‐liquid two‐phase region. The EOS and data closely agree on where the Hugoniot passes through the two‐phase region, which corresponds to where aluminum melts. The bulk sound speed in the crystal region is consistent with the data, given the uncertainty in the experimental procedure. We also estimate shear moduli by using the experimental longitudinal sound speed data and the calculated bulk modulus. The shear modulus satisfies the approximation GS/BS=constant, within experimental error bars, throughout the crystal region on the Hugoniot.

Liquid-state properties from first-principles density functional theory calculations: Static properties

In order to test the Vibration-Transit (V-T) theory of liquid dynamics, ab initio density functional theory (DFT) calculations of thermodynamic properties of Na and Cu are performed and compared with experimental data. The calculations are done for the crystal at T = 0 and T_m, and for the liquid at T_m. The key theoretical quantities for crystal and liquid are the structural potential and the dynamical matrix, both as function of volume. The theoretical equations are presented, as well as details of the DFT computations. The properties compared with experiment are the equilibrium volume, the isothermal bulk modulus, the internal energy and the entropy. The agreement of theory with experiment is uniformly good. Our primary conclusion is that the application of DFT to V-T theory is feasible, and the resulting liquid calculations achieve the same level of accuracy as does ab initio lattice dynamics for crystals. Moreover, given the well established reliability of DFT, the present results provide a significant confirmation of V-T theory itself.

Mean-atom-trajectory model for the velocity autocorrelation function of monatomic liquids.

We present a model for the motion of an average atom in a liquid or supercooled liquid state and apply it to calculations of the velocity autocorrelation function Z(t) and diffusion coefficient D. The model trajectory consists of oscillations at a distribution of frequencies characteristic of the normal modes of a single potential valley, interspersed with position- and velocity-conserving transits to similar adjacent valleys. The resulting predictions for Z(t) and D agree remarkably well with molecular dynamics simulations of Na at up to almost three times its melting temperature. Two independent processes in the model relax velocity autocorrelations: (a) dephasing due to the presence of many frequency components, which operates at all temperatures but which produces no diffusion, and (b) the transit process, which increases with increasing temperature and which produces diffusion. Because the model provides a single-atom trajectory in real space and time, including transits, it may be used to calculate all single-atom correlation functions.

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.

Improved model for the transit entropy of monatomic liquids

In the original formulation of vibration-transit (V-T) theory for monatomic liquid dynamics, the transit contribution to entropy was taken to be a universal constant, calibrated to the constant-volume entropy of melting. This model suffers two deficiencies: (a) it does not account for experimental entropy differences of +/-2% among elemental liquids and (b) it implies a value of zero for the transit contribution to internal energy. The purpose of this paper is to correct these deficiencies. To this end, the V-T equation for entropy is fitted to an overall accuracy of +/-0.1% to the available experimental high-temperature entropy data for elemental liquids. The theory contains two nuclear motion contributions: (a) the dominant vibrational contribution S_{vib}(T/theta_{0}) , where T is temperature and theta_{0} is the vibrational characteristic temperature, and (b) the transit contribution S_{tr}(T/theta_{tr}) , where theta_{tr} is a scaling temperature for each liquid. The appearance of a common functional form of S_{tr} for all the liquids studied is a property of the experimental data, when analyzed via the V-T formula. The resulting S_{tr} implies the correct transit contribution to internal energy. The theoretical entropy of melting is derived in a single formula applying to normal and anomalous melting alike. An ab initio calculation of theta_{0} , based on density-functional theory, is reported for liquid Na and Cu. Comparison of these calculations with the above analysis of experimental entropy data provides verification of V-T theory. In view of the present results, techniques currently being applied in ab initio simulations of liquid properties can be employed to advantage in the further testing and development of V-T theory.

Estimating the Viscosity Coefficient of Liquid Metals from Vibration‐Transit Theory

We describe and test a simple method for calculating the viscosity coefficient η for liquid metals over a range of densities and temperatures. The method uses a model of atomic motion in a liquid based on the vibration‐transit (V‐T) theory of liquid dynamics to calculate the self‐diffusion coefficient D, and then uses D and the Stokes‐Einstein relation to compute η. We consider the accuracy of both the V‐T model for D and the Stokes‐Einstein relation; we then compute η for 21 liquid metals at melt at 1 bar, finding that our results agree with experiment to 18% accuracy. This is somewhat more accurate than other empirical methods and not much less accurate than first principles molecular dynamics calculations, while being substantially less computationally intensive than the latter.

Systematics of Compaction for Porous Metal and Metal-Oxide Systems

The effects of particle morphology and initial density is examined with respect to the shock densification response of initially porous metal (Cu) and metal-oxide (CeO2) materials. Specifically, the ability of a continuum-level compaction model to capture the measured densification trends as a function of initial density and particle morphology are investigated. Particle morphology is observed to have little effect on the densification response of both Cu and CeO2, while initial density appears to have a stronger effect. In terms of continuum-level compaction strength, Cu and CeO2 exhibit dissimilar trends.

V-T theory for the Self-Intermediate Scattering Function in a Monatomic Liquid

In V-T theory the atomic motion is harmonic vibrations in a liquid-specific potential energy valley, plus transits, which move the system rapidly among the multitude of such valleys. In its first application to the self intermediate scattering function (SISF), V-T theory produced an accurate account of molecular dynamics (MD) data at all wave numbers q and time t. Recently, analysis of the mean square displacement (MSD) resolved a crossover behavior that was not observed in the SISF study. Our purpose here is to apply the more accurate MSD calibration to the SISF, and assess the results. We derive and discuss the theoretical equations for vibrational and transit contributions to the SISF. The time evolution is divided into three successive intervals: the vibrational interval when the vibrational contribution alone accurately accounts for the MD data; the crossover when the vibrational contribution saturates and the transit contribution becomes resolved; and the diffusive interval when the transit contribution alone accurately accounts for the MD data. The resulting theoretical error is extremely small at all q and t. V-T theory is compared to mode-coupling theories for the MSD and SISF, and to recent developments in Brownian motion experiments and theory.