Force Correlation Function Research Articles

Ionic dynamics in computer simulated molten LiNO3. I. Translational and reorientational motion

In order to study the microdynamics of translational and orientational motion of ions in liquid electrolytes, a molecular dynamics simulation of molten LiNO3, which includes all degrees of freedom of vibrating nitrate ions, has been performed. Simple Coulomb pair potentials with Born-type repulsions have been adopted in the simulation. The calculated properties include: translational velocity and force correlation functions, rotational correlation functions, and angular momentum and torque correlation functions. The nature of the ionic motion is analyzed in terms of quasioscillation and quasilibration. Local structures surrounding nitrate ions, which can be characterized by, e.g., the coordination number, are found to persist for a long time (∼1 ps). The coupling between the environmental structure and the translational and orientational behaviors of a given nitrate ion has been examined through selectively sampled correlation functions. We found that the change of coordination number enhances the translational diffusion and the rotational diffusion is slow if the coordination number remains low. This shows that the dynamics are very sensitive to the exact shape of the potential and the dynamics itself. Results of the simulation are compared with the experimental diffusion constants and dynamical behaviors revealed by Raman spectroscopy. While the interionic interaction assumed has been found to describe quite well the static structure of molten LiNO3, simulated translational diffusion and rotational relaxation seem to be somewhat too fast, suggesting too shallow an attractive well in the potential we have assumed.

Liquid Structural Theories Related via a Force Correlation Function

Abstract Four approximate theories of dense liquid structure are cast into directly comparable form using a force correlation function. Numerical comparisons via computer simulation data are proposed as a result.

Time behavior of the correlation functions in a simple dissipative quantum model

The force and velocity correlation functions for a particle interacting with a bath are calculated within a model allowing for finite memory effects. The relevance of a Brownian picture is delineated in view of the respective behavior of these functions and appears fully inadequate below some cross-over temperature; then, the interplay between quantum and thermal fluctuations yields some long time tails on the same time scale for both correlation functions. The real space transient diffusion coefficient is found to exceed its asymptotic Einstein value for most times in that regime. The limiting case of an infinitely short memory time is also investigated and is seen to produce weak divergences on a time scale which is small as compared to the other characteristic times.

Photon correlation spectroscopy of ordered colloidal systems

A time-dependent harmonic potential and a memory function formulation have been used for the dynamics of the interacting Brownian particles in ordered colloidal systems. Approximations have been introduced at the level of the force correlation function. The theoretical scattering functions agree well with the photon correlation spectroscopy data for polystyrene colloidal dispersions in aqueous environments.

Brownian dynamics simulation of interacting particles

The dynamics of the ions in an electrical double layer has been studied using stochastic dynamics simulation methods. The theory of a brownian particle in an external force field is discussed and a modified form of the fluctuation-dissipation theorem is derived. With slowly varying external forces the simulations can be restricted to configuration space and the correction term to the fluctuation-dissipation theorem, arising from the external force, can be neglected. It is shown that the diffusion coefficient of a brownian particle can be calculated from the force correlation function. In comparison with calculations from mean-square displacements the former way turns out to be much more efficient. Finally, a relation between the external force () and the random force () is derived such that = -2 .

Dynamics of interacting Brownian particles

A memory function formulation has been used for the dynamics of interacting Brownian particles in colloidal systems and polymer dispersions. By introducing approximations at the level of the force correlation function, the theoretical scattering functions agree well with the experimental photon correlation spectroscopy data.

The Electrical Resistivity of Clustering Alloys

AbstractThe electrical resistivity ϱ of clustering alloys is calculated using the force–force correlation function scheme in the weak‐scattering limit. In the high‐temperature region both signs of the extra‐resistivity Δϱ caused by clustering effects are obtained to be possible in dependence on the Fermi level. At temperatures below the critical temperature of the alloy, where a cluster‐growing process occurs, the resistivity is calculated in dependence on the cluster size. For such alloys, where clustering causes a positive extra‐resistivity Δϱ in the high‐temperature region, the well‐known maximum of the resistivity at a critical cluster size is obtained for spherical clusters. The resistivity of alloys with a negative Δϱ in the high‐temperature region decreases monotonically during a growing process of spherical clusters. Some further numerical results for elliptical and plane clusters are given.

Force and torque correlations in diatomic fluids

A simple relationship is derived between the mean squared force and mean squared torque in homonuclear diatomic fluids which successfully accounts for published simulation data spanning a wide range of bond lengths and densities in the liquid range for molecules without quadrupole moments. Modifications required in heteronuclear systems are discussed and found to be consistent with the scant information available in this area. The expression for homonuclear systems can be extended to explain the relation between the force and torque time correlation functions in these and triatomic fluids. Finally, the results are used to link the parameter κ, introduced by Kivelson et al., to the slip limit friction coefficient for molecular reorientation.

A statistical mechanical theory of the self-diffusion coefficient of simple ions in electrolyte solutions

A statistical mechanical theory of the self-diffusion coefficient of ions in solutions of simple electrolytes has been developed. Beginning with a generalized Langevin equation the self-diffusion coefficients of ions may be evaluated at the zero-frequency limit of the Laplace transform of the random force correlation function. We assume that the random force acting on the tagged ion may be separated into contributions from the solvent part, due to the surrounding solvent molecules and an ionic part due to all the other ions. Further, we assume that the evolution of the ionic random force is governed by the Smoluchowski operator. With these assumptions and using the Debye–Hückel pair correlation function, the Onsager limiting law may be derived. Numerical calculations using the HNC pair correlation function shows that our theory can describe experimental data of moderately concentrated solutions adequately.

A simplified approach to kinetic theory of dense fluids

Time correlation functions may be related to the fluctuating force correlation function in the sub-ensemble with fixed initial particle velocity. The latter may be evaluated by following the trajectories of particle pairs in the fluid.

Boltzmann equation and force–force correlation function for electrical resistivity

AbstractThe frequency‐dependent electrical resistivity is investigated through the Boltzmann equation and the Kubo formalism in the low‐frequency weak‐scattering regime where the Boltzmann equation is known to be valid. A commonly used force‐force correlation‐function formula for dc resistivity is shown to be derivable from the Boltzmann equation as well as from the Kubo formalism. The derivation assumes that expansions used in the ωτ ≫ I regime for the “memory function” or for the inverse of the relaxation time τ are valid in the limit as the frequency ω goes to zero. This procedure and the correlation‐function formula are shown to give incorrect results even for the zero‐temperature case if the scattering is anisotropic, i.e., if τ varies over the Fermi surface. Thus the previous criticism of the force‐force correlation function is extended to the anisotropic scattering regime, and the direct connection between this function and the Boltzmann equation is shown.

Time correlation functions in classical dense fluids

The velocity autocorrelation functions and memory functions of dense classical fluids may be directly obtained from the static radial distribution function g(r) in an approximate way. Following the Mori projection operator formalism, the memory functions may be related to the fluctuating force correlation. At low densities, these functions may be evaluated by following the trajectories of particle pairs in the interatomic potential. At higher densities, the force correlation functions can be evaluated approximately from particle pair trajectories via the potential of the mean force. The contributions to the memory function come mainly from particle pairs with rather specific and rather short interatomic distances. At higher temperatures, this specific distance is even shorter, hence the memory function decays quickly with time. At lower temperatures, a negative region of the memory function may develop. On the other hand, there is relatively little density dependence of the normalized memory function. The results for argon fluids at various densities and temperatures agree satisfactorily with the molecular dynamics and the Enskog values. The decrease of the diffusion coefficient with density is partly due to the nature of g(r) which reflects the stronger clustering of atoms at higher densities.

Vibrational relaxation in the weak coupling limit: The role of molecular symmetry

We present a set of simple symmetry rules for atomic force correlation functions which determine vibrational relaxation rates in liquids and gases. In addition to greatly simplifying the rate expressions, these rules may provide a basis for understanding vibrational propensity rules in the weak coupling limit.

Vibrational energy relaxation in liquids

A theory of vibrational relaxation of polyatomic molecules in liquids is presented which expresses the rate of decay of a vibrational state in terms of atomic force autocorrelation functions. These force correlation functions can be estimated from hydrodynamics or from an independent binary collision model. The results of calculations on small molecules dissolved in inert, cryogenic solvents are discussed, including the dependence of the rates on the details of the intramolecular potential surface.

On the spin‐disorder resistivity of amorphous ferromagnets

AbstractThe spin‐disorder resistivity of amorphous ferromagnets is investigated in the temperature regimes 0 ≦ T ≫ TC and T ≫ TC. The resistivity is calculated within the force–force correlation function method up to the second order in the sd‐exchange constant. A simple structural model for a structurally disordered Heisenberg ferromagnet is presented. At lowest temperatures the spin‐disorder resistivity varies as T3/2 for a sufficient high degree of disorder. It is concluded that the experimentally observed T2‐dependence at these temperatures is of non magnetic origin. This dependence can be qualitatively understood if the thermal smearing of the structure factor is taken into account.

Force–force correlation function approach to the electrical resistivity of disordered transition metal alloys

AbstractThe method of the force‐force correlation functions is used to derive, beyond the weak scattering approximation, a tractable approximative formula for the electrical resistivity of disordered binary alloys. The starting point is a one‐particle expression for the resistivity formulated in terms of the total scattering operator T. The local approximation is introduced by replacing the T operator by the atomic t matrices in order to treat transition metal alloys described by muffin‐tin single‐particle Hamiltonians. Short‐range multiple‐scattering corrections to the local approximation are evaluated using a non‐selfconsistent single‐site average‐t‐matrix approximation. Explicit results are given and discussed for the resistivity of a one‐dimensional muffin‐tin model for transition‐metal‐like alloys with resonant states.

Average force and force correlation formulas for momentum transfer cross section

Gerjuoy has related the momentum transfer cross section to the average force. The force correlation function formula of Rousseau, Stoddart, and March is here shown to be transformable into Gerjouy’s result for a spherically symmetrical scatterer.

Modified Fokker-Planck equations

The exact Fokker-Planck-like equation of motion for the singlet phase space density derived by Lebowitz and Resibois has been investigated with the objective of improving on the usual replacement of the exact translational collision operator by the Fokker-Planck operator. The exact collision operator is expressed in terms of a projected force correlation operator which acts on the position and momentum of the tagged particle. The lowest order operator dependence of this correlation function is exposed, resulting in an improved FP collision operator that contains third and fourth-order momentum derivatives and a streaming term, which, roughly speaking, allows free translation during the correlation time of the fluctuating forces. The improved FP operator is used to obtain a short time expansion (up to t 4) for the velocity correlation function. Through a cumulant expansion technique, the K-space singlet density is also obtained and a criteria, based on force correlation functions, is devised for the existence of the super Burnett coefficient.

Force‐correlation function method applied to the electrical resistivity of many‐band metals

AbstractThe method of calculating the electrical resistivity by force–force correlation functions is generalized to many‐band metals. For two‐band metals a formula is derived from which the electrical resistivity R can be calculated by means of a simple perturbation treatment. For the phonon assisted s‐d scattering the temperature dependence of R is calculated for the case that the d‐electrons do not remain in thermal equilibrium. The two‐current formula for the electrical resistivity of ferromagnetic metals and alloys is derived.

Molecular theory of solvated ion dynamics

We present a simple molecular approach for understanding the dynamics of a single ion in a polar solvent. The approach is based on a heuristic development starting from a resolution of the force–force correlation function of a Brownian ion, into parts depending on the attractive ion–solvent interaction and a repulsive, hard core ion–solvent interaction. We show that in the limit of infinitely weak, long range attractive ion–solvent interactions the molecular theory reduces to the phenomenological dielectric friction picture for the drag on an ion. The molecular theory also gives sensible results in the case of strong, short range attractive interactions. In this case the theory essentially indicates that the drag on the ion reduces to the drag on a composite body, i.e., a solventberg consisting of the ion and the molecules in the first solvation shell.