Van Hove Function Research Articles

MD study of the mixed alkali effect in a lithium&.zsbnd;potassium metasilicate glass

A molecular dynamics simulation of (Li,K)SiO3 has been carried out to determine the mixed alkali effect. A decrease in dynamic property by mixing is reproduced by the simulation. The distinct part of the van Hove function reveals that the jump of alkali ions into a site previously occupied by a different type of alkali ion has a low probability in a mixed alkali glass. The result means that an interception of the mutual diffusion pathways occurs in the glassy state.

Structure and dynamics of yukawa systems

Results of molecular dynamics simulations modelling two component charge stabilized colloidal particles interacting via a Yukawa potential are presented. After cooling, the systems freeze into either substitutionally disordered imperfect crystals or into glasslike states. This freezing is characterized by the divergence of a suitable correlation time due to loss of ergodicity. Describing the structure by bond correlation functions, local orientational ordering is observed in the glassy states which is not present in the liquid. In the liquid the diffusion constant obeys an Arrhenius law. As can be deduced from the van Hove functions, in the crystal the particles only oscillate around their mean positions, whereas in the glassy state some particles move by occupying neighbouring sites during local rearrangements.

Interaction-site representation of the Smoluchowski–Vlasov equation: The space–time correlation functions in a molecular liquid

A site–site representation of the Smoluchowski–Vlasov equation is proposed based on the generalized Langevin equation, and on the site–site Ornstein–Zernike equation for the molecular liquid. The method is applied for calculating the space–time pair correlation function proposed by van Hove. The results are compared with a molecular dynamics simulation of water. The site–site Smoluchowski–Vlasov equation predicts most of the essential features of the van Hove function for water.

Correlations in the motion of atoms in liquid silicon.

A first-principles molecular-dynamics study of atomic dynamical correlations in liquid silicon is presented. The properties studied here include the van Hove correlation function, velocity autocorrelation function, spectral density, and non-Gaussian corrections to the incoherent van Hove function. It is shown that these correlation functions exhibit an oscillatory behavior at short and intermediate times due to stretching vibrations of covalent bonds that form in the metallic liquid. Quantitative estimates are given for the time scale of typical ionic relaxation times and for the self-diffusion coefficient.

Dielectric relaxation of electrolyte solutions

The dielectric relaxation theory of electrolyte solutions is formulated in terms of solvent–solvent, ion–ion, and ion–solvent van Hove time correlation functions. General wave vector frequency-dependent expressions are given for the longitudinal components of the relevant (i.e., polarization–polarization, current–current, current–polarization, polarization–current) time correlation functions and of the susceptibility, dielectric, and conductivity tensors. The Kerr theory relating the distinct and self parts of the van Hove functions is extended to mixtures of molecular fluids and solved explicitly in the k→0 limit for solutions of spherical ions (assuming that the self part of the van Hove functions is given by Fick’s law) immersed in polar solvents. At this level of theory, the van Hove functions, the time correlation functions and the susceptibilities are all found to depend upon coupled ion–solvent motion. However, the dynamical coupling terms are shown to cancel exactly in the final expressions for the conductivity and dielectric constant yielding relatively simple results. Specifically, the conductivity obtained is independent of frequency and is related to the self diffusion constants of the ions by the Nernst–Einstein expression. If a spherical diffusor model is chosen for the solvent, then the frequency-dependent dielectric constant is given by a Debye-type formula with a concentration dependent relationship connecting the Debye and self reorientational relaxation times of the solvent.These results are discussed in the context of previous theories and experimental observations. It is shown that, although obviously oversimplified, the present theory does qualitatively predict the correct concentration dependence of the observed relaxation times for a number of salt solutions.

Dielectric relaxation of liquid mixtures

General expressions in terms of van Hove time correlation functions are given for the wave vector frequency-dependent dielectric function of multicomponent mixtures. The van Hove functions are obtained by applying the Kerr approximation and the dielectric relaxation at zero wave vector is considered in detail. At this level of theory, the frequency-dependent dielectric constant depends upon the self-reorientational correlation times of the various species involved and upon the equilibrium pair correlation functions. It is shown that if the self-correlation times are assumed to be given by the Stokes–Debye relationship, and if the equilibrium direct correlation functions obey certain relatively weak conditions, then for particles of equal size (i.e., the self-correlation times are the same for all species) the dielectric relaxation behavior can be described by a simple Debye formula with a single concentration-dependent relaxation time. This observation is independent of the number of components, of the concentration, and of the molecular dipole moments of the different species present. It may help explain why for some binary mixtures of polar molecules experimental measurements indicate only a single relaxation channel. The exact Kerr result for binary mixtures is expressed explicitly as the sum of two Lorentzians, and some numerical results are given for solutions of dipolar hard spheres of different diameter.

Dynamics of molecular liquids: A comparison of different theories with application to wave vector dependent dielectric relaxation and ion solvation

Explicit expressions are given for the Fourier–Laplace transform of the van Hove function for fluids of particles interacting through orientation dependent pair potentials. The expressions are obtained from the Kerr approximation together with diffusion models for the self part of the van Hove function and apply at all frequencies and wavelengths. Both spherical (scalar diffusion constants) and nonspherical (tensorial diffusion constants) diffusors are considered and the theory is applied to k-dependent dielectric relaxation and ion solvation dynamics. The required equilibrium structure is obtained using the hypernetted chain (HNC) or reference hypernetted chain (RHNC) theories and Kerr/RHNC(HNC) results are given for fluids of dipolar hard spheres, dipolar hard ellipsoids, and water-like particles. Comparisons are made with earlier work based upon coupling equilibrium theories, such as the mean spherical approximation (MSA) for dipolar hard spheres with a dynamical equation of the Smoluchowski–Vlasov (SV) type. It is shown that for spherical diffusors with the self part of the van Hove function treated at the Fick–Debye level, the SV and Kerr equations are in fact equivalent. However, even for spherical diffusors, the results obtained can differ substantially depending upon the equilibrium theory and/or the molecular model employed. For nonspherical diffusors, the anisotropy of the rotational diffusion tensor can also be an important parameter influencing the k-dependent dielectric relaxation.

Rotational motion in molecular liquids

This paper is concerned with rotational motion and the coupling between rotational and translational motion in molecular fluids. The van Hove angular space–time correlation function is expanded in a basis set of rotational invariants and some properties of the expansion are discussed. General expressions for collective reorientational time correlation functions and relaxation times are given in terms of the expansion coefficients. It is shown that this expansion allows the Kerr and Vineyard approximations which relate the distinct and self parts of the van Hove function to be cast in tractable form and Fourier–Laplace space solutions can be obtained for all frequencies and wave vectors. General solutions are given as well as explicit results for the simpler but important k=0 case. The dielectric relaxation of polar–nonpolarizable molecules is discussed and both infinite samples and the reaction field boundary conditions sometimes employed in the computer simulation of polar fluids are considered. It is shown that in the Kerr theory the frequency dependent dielectric constant ε(ω) is correctly independent of the boundary conditions applied. This is not true of the Vineyard approximation. Applying the Kerr theory and using a result for the self part of the van Hove function recently given by Caillol, we investigate the dielectric relaxation of both spherical and nonspherical diffusors. The nonspherical model is that of Berne and Pecora and couples the rotational and translational diffusion. It is found that within the Kerr approximation spherical diffusors behave as Debye dielectrics at all frequencies. However, in the nonspherical case ε(ω) can deviate substantially from the Debye result and we conclude that for highly anisotropic diffusors the coupling between rotational and translational motion can significantly contribute to the observed dielectric relaxation.

A computer simulation study for model liquid ammonia—time correlation functions

Time correlation functions for liquid ammonia using an intermolecular potential model developed recently (Mansour and Murad, 1987) are reported. The potential model which consists of a Lennard-Jones central part, a point dipole and quadrupole and dipole polarizability (multi-body) has been found (Mansour and Murad 1987) to give an accurate representation of many measured thermodynamic and transport properties. The properties reported here include autocorrelation functions and correlation times for translational velocity, rotational velocity, force, torque, two re-orientational functions, and the self and distinct van Hove functions. Where possible the results obtained have been compared with experimental results.

Smoluchowski fluctuation theory of ion transport in solutions

We developed a Smoluchowski-level theory of time-dependent ion concentration fluctuations at equilibrium, including higher-order correlations, but with hydrodynamic interactions suppressed. A partial sum over the higher-order correlations, a chain sum, eliminates divergences. From the resulting renormalized self-van Hove function we recover Onsager’s limiting law for the self-diffusion coefficients of the ions. The limiting law comes from the chain sums that are neglected in a ‘‘linearization’’ approximation used in some earlier work. The corresponding development for the distinct van Hove functions is merely indicated. We apply these results to a solution of a single spherical polyion with many small ions, to obtain the self-diffusion coefficient for the polyion in near-limiting law conditions, and to simple models for aqueous NaCl and CuSO4 solutions, to calculate the self-diffusion coefficients of the ions and the Maxwell (Debye–Falkenhagen) relaxation, all in the concentration range up to 1 M.

The roto-translational diffusion of a molecule in a fluid

The coupling between the translational and rotational motions of a molecule in a fluid is investigated with the help of a diffusion equation for the self van Hove function G s. This equation is obtained from an exact balance equation for the microscopic density of molecules by introducing in the hydrodynamic limit phenomenological tensorial diffusion coefficients. Then substituting in this equation the expansion of G s on a complete basis set of rotational invariants we obtain an infinite system of coupled diffusion equations for the coefficients of the G s expansion. From the solution at wave vector k = 0 of this system we derive a new expression for the reorientational correlation times of a molecule.

Fractal dimensionality and microscopic analysis of the dynamics in simple liquids

The idea to evaluate the «average fractal dimension» of the motion in a condensed phase system, on the basis of the analysis of the fractal properties of the trajectories of its microscopic components has been further developed. The fractal dimension of particle trajectories, evaluated through the correlation density integral, is formally related with the self-part of the dynamic structure factorGs(r, t) (the self-part of the Van Hove function); so far a bridge between fractal and thermodynamical properties has been built up.

A Vineyard-type approximation for the distinct term of the dynamic pair correlation function

The distinct term of the van Hove function G(r, t) is approximated by a Vineyard type convolution approximation. The convolution is made with a certain “self” correlation function G sd (r, t) generated by dynamics of pairs of particles. This G sd (r, t) function accounts partly for cross correlation and thus improves the results achievable by the convolution with the pure self term of G(r, t). The results computed by the present method are compared with molecular dynamics data at two liquid-like densities. Good agreement is found for the intermediate range of wave vectors. The approximation is recommended, in particular, for purposes of Fourier transformation.

Computation and analysis of the dynamic structure factor S(k, ω) for small wave vectors

Molecular dynamics (MD) calculations have been performed for the Lennard-Jones (12-6) potential function using 2048 particles. Using conventional parameters the results may be compared with those for liquid argon. The dynamic structure factor S(k, ω) has been determined both by Fourier inversion of the intermediate scattering function F(k, t) and from the longitudinal current-current correlation function C ∥(k, t). Particular attention was paid to the recurrence time of the system. The results for S(k, ω) by the two methods agree within 5 per cent for the whole region of small k-vectors considered. Double Fourier inversion of the van Hove function G(r, t) led to insufficiently accurate results for these small k-values. In view of the present data, the MD-results of Levesque et al. [1] for S(k, ω) have only a qualitative character. These latter data appear to contain truncation errors due to incomplete Fourier transformations. Using a hydrodynamic assumption for F(k, t) we were able to extract the transpor...

Neutron-photon studies of condensed matter

A theory of simultaneous photon absorption and inelastic neutron scattering is developed by treating the photon and neutron-matter interactions perturbatively. The leading-order mixing between the interactions shows that the neutron scattering cross-section is proportional to the dynamic structure factor (or Van Hove function) evaluated at an energy that is enhanced by the photon energy. The photon induced modification of the scattering vector is negligible. Thus, the proposed technique affords the possibility of measuring the dynamic structure factor at large energies and modest wavevectors which is a domain that is usually difficult to access because of kinematic constraints. The theory is developed in detail for some models of nuclear and magnetic systems. The results show that, in most cases, the experiments are likely to demand the use of very high intensity light sources. A particularly promising application appears to be in the study of electron plasmas since, using readily available pulsed lasers, the neutron cross-section is comparable with that for pure magnetic scattering.

The translational motion of molecules at short times: a study using molecular dynamic simulations and the model of itinerant oscillation

Abstract A newly developed version of the itinerant oscillator model for molecular translation in fluids is evaluated by means of data obtained from a molecular dynamics simulation of the fluid using an atom-atom potential. The data consists of: 1. Velocity ( v ) and force ( F ) autocorrelation functions. 2. Mean square displacements of molecular centres of mass and even moments thereof. 3. Autocorrelation of v2n and F2n (as a means of investigating the deviation of v and F from Gaussian behaviour). 4. The van Hove function Gs(R,t). The latter is matched against incoherent neutron scattering data observed by Dasannacharya and Rao [13] for liquid argon. The theoretical picture is realistic except that the neglect of cross-correlations and rotation-translation coupling is clearly affecting the agreement between the simulated and calculated mean square displacements. The self van Hove function Gs( R ,t) exhibits complicated non-Gaussian behaviour which will be difficult to follow using the theory of Brownian motion.

Mode coupling theory of charge fluctuation spectrum in a binary ionic liquid

The paper reports a mode coupling calculation of the memory function for the dynamics of long-wave-length charge fluctuations in a symmetric binary ionic liquid, as a simple model for molten alkali halides. Symmetry under charge conjugation is exploited to derive selection rules for the decay of single-mode excitations into multimode excitations. An estimation of the two-mode decay channels indicates that they can be approximately separated into fast processes and slow processes. The former involve a pair of outcoming longitudinal modes, one of acoustic character and the other of optical character, and are the main cause of broadening of the high-frequency peak in the van Hove structure factor for charge density fluctuations. The slow-decay channels involve either a longitudinal-optic and a transverse-acoustic mode or a longitudinal-acoustic and a transverse-optic mode and are dominant in the low-frequency region of the spectrum where Drude-lile conduction is occurring. The calculated shape of the van Hove function is in reasonable agreement with the available computer simulation data, after an empiricla scaling of the spectrum of fluctuating forces.

Non-Markovian probability density functions with applications to itinerant oscillator models

Starting from the linear Mori-Kubo generalized Langevin equation, conditional probability density functions are derived for independent dynamical variables of a general system under the action of (non-markovian) gaussian force or torque. The special techniques developed by Adelman (1976, 1977) for brownian oscillators are here extended to the general case. As applications of the general theory, angular momentum and orientational probability density functions are computed for models of molecular rotation in fluids. The rotational dynamics of each molecule is described by an itinerant oscillator/librator. The periodicity of spatial orientation is treated in terms of wrapped distributions, and hence the planar-reorientational counterpart of the translational self van Hove function is derived. It is shown that both momentum and orientational probability density functions can exhibit widely varying time-decay properties, which are directly interpretable in terms of the structure and dynamics of the molecular f...

Atomic Dynamics in Liquids

The subject of dynamics of electrons and atoms in liquids is presented in a form which should be useful for established research workers in the field of the physics and chemistry of the liquid state. Topics covered include: static structure and thermodynamics; calculation of liquid structure from a law of force; liquid dynamics and time dependent correlation functions; hydrodynamic forms of correlation functions and generalized hydrodynamics; microscopic theories of the van Hove function; structure and dynamics of binary fluids; charged fluids; helium fluids; critical phenomena; the liquid surface. (GHT)

Kinetic equations for dense fluids

It is demonstrated that the kinetic equation of Davis's effective potential theory follows directly from the application of well-defined approximations to the three-body correlations involved in the second equation of the BBGKY hierarchy. The same, simple mathematical techniques involved in this demonstration are used to derive two other kinetic equations, one of which is a generalization to high densities of the Boltzmann equation. In order to facilitate its application to the calculation of the van Hove and other correlation functions, the kinetic equation of the effective potential theory is Fourier-Laplace transformed: explicit formulae are given for the matrix elements of all operators that occur in this equation.