Fast decay of the velocity autocorrelation function in dense shear flow of inelastic hard spheres

We find in complementary experiments and event-driven simulations of sheared inelastic hard spheres that the velocity autocorrelation function psi(t) decays much faster than t(-3/2) obtained for a fluid of elastic spheres at equilibrium. Particle displacements are measured in experiments inside a gravity-driven flow sheared by a rough wall. The average packing fraction obtained in the experiments is 0.59, and the packing fraction in the simulations is varied between 0.5 and 0.59. The motion is observed to be diffusive over long times except in experiments where there is layering of particles parallel to boundaries, and diffusion is inhibited between layers. Regardless, a rapid decay of psi(t) is observed, indicating that this is a feature of the sheared dissipative fluid, and is independent of the details of the relative particle arrangements. An important implication of our study is that the non-analytic contribution to the shear stress may not be present in a sheared inelastic fluid, leading to a wider range of applicability of kinetic theory approaches to dense granular matter.

Dynamics of a dilute sheared inelastic fluid. I. Hydrodynamic modes and velocity correlation functions

The hydrodynamic modes and the velocity autocorrelation functions for a dilute sheared inelastic fluid are analyzed using an expansion in the parameter epsilon=(1-e)(1/2), where e is the coefficient of restitution. It is shown that the hydrodynamic modes for a sheared inelastic fluid are very different from those for an elastic fluid in the long-wave limit, since energy is not a conserved variable when the wavelength of perturbations is larger than the ``conduction length.'' In an inelastic fluid under shear, there are three coupled modes, the mass and the momenta in the plane of shear, which have a decay rate proportional to k(2/3) in the limit k -> 0, if the wave vector has a component along the flow direction. When the wave vector is aligned along the gradient-vorticity plane, we find that the scaling of the growth rate is similar to that for an elastic fluid. The Fourier transforms of the velocity autocorrelation functions are calculated for a steady shear flow correct to leading order in an expansion in epsilon. The time dependence of the autocorrelation function in the long-time limit is obtained by estimating the integral of the Fourier transform over wave number space. It is found that the autocorrelation functions for the velocity in the flow and gradient directions decay proportional to t(-5/2) in two dimensions and t(-15/4) in three dimensions. In the vorticity direction, the decay of the autocorrelation function is proportional to t(-3) in two dimensions and t(-7/2) in three dimensions.

Subgrid scale fluid velocity timescales seen by inertial particles in large-eddy simulation of particle-laden turbulence

Large-eddy simulation (LES) has emerged as a promising tool for simulating turbulent flows in general and, in recent years,has also been applied to the particle-laden turbulence with some success (Kassinos et al., 2007). The motion of inertial particles is much more complicated than fluid elements, and therefore, LES of turbulent flow laden with inertial particles encounters new challenges. In the conventional LES, only large-scale eddies are explicitly resolved and the effects of unresolved, small or subgrid scale (SGS) eddies on the large-scale eddies are modeled. The SGS turbulent flow field is not available. The effects of SGS turbulent velocity field on particle motion have been studied by Wang and Squires (1996), Armenio et al. (1999), Yamamoto et al. (2001), Shotorban and Mashayek (2006a,b), Fede and Simonin (2006), Berrouk et al. (2007), Bini and Jones (2008), and Pozorski and Apte (2009), amongst others. One contemporary method to include the effects of SGS eddies on inertial particle motions is to introduce a stochastic differential equation (SDE), that is, a Langevin stochastic equation to model the SGS fluid velocity seen by inertial particles (Fede et al., 2006; Shotorban and Mashayek, 2006a; Shotorban and Mashayek, 2006b; Berrouk et al., 2007; Bini and Jones, 2008; Pozorski and Apte, 2009).However, the accuracy of such a Langevin equation model depends primarily on the prescription of the SGS fluid velocity autocorrelation time seen by an inertial particle or the inertial particle–SGS eddy interaction timescale (denoted by $\delt T_{Lp}$ and a second model constant in the diffusion term which controls the intensity of the random force received by an inertial particle (denoted by C_0, see Eq. (7)). From the theoretical point of view, dTLp differs significantly from the Lagrangian fluid velocity correlation time (Reeks, 1977; Wang and Stock, 1993), and this carries the essential nonlinearity in the statistical modeling of particle motion. dTLp and C0 may depend on the filter width and particle Stokes number even for a given turbulent flow. In previous studies, dTLp is modeled either by the fluid SGS Lagrangian timescale (Fede et al., 2006; Shotorban and Mashayek, 2006b; Pozorski and Apte, 2009; Bini and Jones, 2008) or by a simple extension of the timescale obtained from the full flow field (Berrouk et al., 2007). In this work, we shall study the subtle and on-monotonic dependence of $\delt T_{Lp}$ on the filter width and particle Stokes number using a flow field obtained from Direct Numerical Simulation (DNS). We then propose an empirical closure model for $\delta T_{Lp}$. Finally, the model is validated against LES of particle-laden turbulence in predicting single-particle statistics such as particle kinetic energy. As a first step, we consider the particle motion under the one-way coupling assumption in isotropic turbulent flow and neglect the gravitational settling effect. The one-way coupling assumption is only valid for low particle mass loading.

Dynamics of a dilute sheared inelastic fluid. II. The effect of correlations

The effect of correlations on the viscosity of a dilute sheared inelastic fluid is analyzed using the ring-kinetic equation for the two-particle correlation function. The leading-order contribution to the stress in an expansion in epsilon=(1-e)(1/2) is calculated, and it is shown that the leading-order viscosity is identical to that obtained from the Green-Kubo formula, provided the stress autocorrelation function in a sheared steady state is used in the Green-Kubo formula. A systemmatic extension of this to higher orders is also formulated, and the higher-order contributions to the stress from the ring-kinetic equation are determined in terms of the terms in the Chapman-Enskog solution for the Boltzmann equation. The series is resummed analytically to obtain a renormalized stress equation. The most dominant contributions to the two-particle correlation function are products of the eigenvectors of the conserved hydrodynamic modes of the two correlated particles. In Part I, it was shown that the long-time tails of the velocity autocorrelation function are not present in a sheared fluid. Using those results, we show that correlations do not cause a divergence in the transport coefficients; the viscosity is not divergent in two dimensions, and the Burnett coefficients are not divergent in three dimensions. The equations for three-particle and higher correlations are analyzed diagrammatically. It is found that the contributions due to the three-particle and higher correlation functions to the renormalized viscosity are smaller than those due to the two-particle distribution function in the limit epsilon -> 0. This implies that the most dominant correlation effects are due to the two-particle correlations.

How does contrasting dependence of impurity-atom diffusivity on the density of host disordered medium arise?

We report results of molecular dynamics investigations into neutral impurity diffusing within an amorphous solid as a function of the size of the diffusant and density of the host amorphous matrix. We find that self diffusivity exhibits an anomalous maximum as a function of the size of the impurity species. An analysis of properties of the impurity atom with maximum diffusivity shows that it is associated with lower mean square force, reduced backscattering of velocity autocorrelation function, near-exponential decay of the intermediate scattering function (as compared to stretched-exponential decay for other sizes of the impurity species) and lower activation energy. These results demonstrate the existence of size-dependent diffusivity maximum in disordered solids. Further, we show that the diffusivity maximum is observed at lower impurity diameters with increase in density. This is explained in terms of the Levitation parameter and the void structure of the amorphous solid. We demonstrate that these results imply contrasting dependence of self diffusivity (D) on the density of the amorphous matrix, p. D increases with p for small sizes of the impurity but shows an increase followed by a decrease for intermediate sizes of the impurity atom. For large sizes of the impurity atom, D decreases with increase in p. These contrasting dependence arises naturally from the existence of Levitation Effect.

A comparative molecular dynamics study of diffusion of n-decane and 3-methyl pentane in Y zeolite

Molecular dynamics simulations are reported on the structure and dynamics of n-decane and 3-methylpentane in zeolite NaY. We have calculated several properties such as the center of mass-center of mass rdf, the end-end distance distribution, bond angle distribution and dihedral angle distribution. We have also analysed trajectory to obtain diffusivity and velocity autocorrelation function (VACF). Surprisingly, the diffusivity of 3-methylpentane which is having larger cross-section perpendicular to the long molecular axis is higher than n-decane at 300 K. Activation energies have been obtained from simulations performed at 200 K, 300 K, 350 K, 400 K and 450 K in the NVE ensemble. These results can be understood in terms of the previously known levitation effect. Arrhenious plot has higher value of slope for n-decane (5 center dot 9 kJ/mol) than 3-methylpentane (3 center dot 7 kJ/mol) in agreement with the prediction of levitation effect.

Two-Phase Thermodynamic Model for Efficient and Accurate Absolute Entropy of Water from Molecular Dynamics Simulations

Presented here is the two-phase thermodynamic (2PT) model for the calculation of energy and entropy of molecular fluids from the trajectory of molecular dynamics (MD) simulations. In this method, the density of state (DoS) functions (including the normal modes of translation, rotation, and intramolecular vibration motions) are determined from the Fourier transform of the corresponding velocity autocorrelation functions. A fluidicity parameter (f), extracted from the thermodynamic state of the system derived from the same MD, is used to partition the translation and rotation modes into a diffusive, gas-like component (with 3Nf degrees of freedom) and a nondiffusive, solid-like component. The thermodynamic properties, including the absolute value of entropy, are then obtained by applying quantum statistics to the solid component and applying hard sphere/rigid rotor thermodynamics to the gas component. The 2PT method produces exact thermodynamic properties of the system in two limiting states: the nondiffusive solid state (where the fluidicity is zero) and the ideal gas state (where the fluidicity becomes unity). We examine the 2PT entropy for various water models (F3C, SPC, SPC/E, TIP3P, and TIP4P-Ew) at ambient conditions and find good agreement with literature results obtained based on other simulation techniques. We also validate the entropy of water in the liquid and vapor phases along the vapor-liquid equilibrium curve from the triple point to the critical point. We show that this method produces converged liquid phase entropy in tens of picoseconds, making it an efficient means for extracting thermodynamic properties from MD simulations.

Temperature and concentration dependence of adsorption properties of methane in NaY: a molecular dynamics study

Molecular dynamics calculations on methane sorbed in NaY (Si/Al = 3.0) employing realistic methane-methane and methane-zeolite intermolecular potential functions at different temperatures (50, 150, 220, and 300 K) and concentrations (2, 4, 6, and 8 molecules/cage) are reported. The thermodynamic results are in agreement with the available experimental data. Guest-guest and guest-host radial distribution functions (rdfs), energy distribution functions, distribution of cage occupancy, center-of-cage-center-of-mass (coc-com) rdfs, velocity autocorrelation functions for com and angular motion and the Fourier transformed power spectra, and diffusion coefficients are presented as a function of temperature and concentration. At 50 K, methane is localized near the adsorption site. Site-site migration and essentially free rotational motion are observed at 150 K. Molecules preferentially occupy the region near the inner surface of the alpha-cage. The vibrational frequencies for the com of methane shift toward higher values with decreasing temperature and increasing adsorbate concentration. The observed frequencies for com motion are 36, 53, and 85 cm-1 and for rotational motion at 50 K, 95 and 150 cm-1 in agreement with neutron scattering data. The diffusion coefficients show a type I behavior as a function of loading in agreement with NMR measurements. Cage-to-cage diffusion is found to be always mediated by the surface.

Entropy and dynamics of water in hydration layers of a bilayer

We compute the entropy and transport properties of water in the hydration layer of dipalmitoylphosphatidylcholine bilayer by using a recently developed theoretical scheme two-phase thermodynamic model, termed as 2PT method; S.-T. Lin et al., J. Chem. Phys. 119, 11792 (2003)] based on the translational and rotational velocity autocorrelation functions and their power spectra. The weights of translational and rotational power spectra shift from higher to lower frequency as one goes from the bilayer interface to the bulk. Water molecules near the bilayer head groups have substantially lower entropy (48.36 J/mol/K) than water molecules in the intermediate region (51.36 J/mol/K), which have again lower entropy than the molecules (60.52 J/mol/K) in bulk. Thus, the entropic contribution to the free energy change (T Delta S) of transferring an interface water molecule to the bulk is 3.65 kJ/mol and of transferring intermediate water to the bulk is 2.75 kJ/mol at 300 K, which is to be compared with 6.03 kJ/mol for melting of ice at 273 K. The translational diffusion of water in the vicinity of the head groups is found to be in a subdiffusive regime and the rotational diffusion constant increases going away from the interface. This behavior is supported by the slower reorientational relaxation of the dipole vector and OH bond vector of interfacial water. The ratio of reorientational relaxation time for Legendre polynomials of order 1 and 2 is approximately 2 for interface, intermediate, and bulk water, indicating the presence of jump dynamics in these water molecules. (C) 2010 American Institute of Physics. doi:10.1063/1.3494115]

Diffusion Anomaly in Silicalite and VPI-5 from Molecular Dynamics Simulations

Molecular dynamics (MD) simulations on rigid and flexible framework models of silicalite and a rigid framework model of the aluminophosphate VPI-5 for different sorbate diameters are reported. The sorbate-host interactions are modeled in terms of simple atom-atom Lennard-Jones interactions. The results suggest that the diffusion coefficient exhibits an anomaly as gamma approaches unity. The MD results confirm the existence of a linear regime for sorbate diameters significantly smaller than the channel diameter and an anomalous regime observed for sorbate diameters comparable to the channel diameter. The power spectra obtained by Fourier transformation of the velocity autocorrelation function indicate that there is an increase in the intensity of the low-frequency component for the velocity component parallel to the direction of motion for the sorbate diameter in the anomalous regime. The present results suggest that the diffusion anomaly is observed irrespective of (1) the geometry and topology of the pore structure and (2) the nature of the host material. The results are compared with the work of Derouane and co-workers, who have suggested the existence of ''floating molecules'' on the basis of earlier theoretical and computational approaches.

Absolute Entropy and Energy of Carbon Dioxide Using the Two-Phase Thermodynamic Model

The two-phase thermodynamic (2PT) model is used to determine the absolute entropy and energy of carbon dioxide over a wide range of conditions from molecular dynamics trajectories. The 2PT method determines the thermodynamic properties by applying the proper statistical mechanical partition function to the normal modes of a fluid. The vibrational density of state (DoS), obtained from the Fourier transform of the velocity autocorrelation function, converges quickly, allowing the free energy, entropy, and other thermodynamic properties to be determined from short 20-ps MD trajectories. The anharmonic effects in the vibrations are accounted for by the broadening of the normal modes into bands from sampling the velocities over the trajectory. The low frequency diffusive modes, which lead to finite DoS at zero frequency, are accounted for by considering the DoS as a superposition of gas-phase and solid-phase components (two phases). The analytical decomposition of the DoS allows for an evaluation of properties contributed by different types of molecular motions. We show that this 2PT analysis leads to accurate predictions of entropy and energy of CO2 over a wide range of conditions (from the triple point to the critical point of both the vapor and the liquid phases along the saturation line). This allows the equation of state of CO2 to be determined, which is limited only by the accuracy of the force field. We also validated that the 2PT entropy agrees with that determined from thermodynamic integration, but 2PT requires only a fraction of the time. A complication for CO2 is that its equilibrium configuration is linear, which would have only two rotational modes, but during the dynamics it is never exactly linear, so that there is a third mode from rotational about the axis. In this work, we show how to treat such linear molecules in the 2PT framework.

Dependence of diffusivity on density and solute diameter in liquid phase: A molecular dynamics study of Lennard-Jones system

Investigations into the variation of self-diffusivity with solute radius, density, and degree of disorder of the host medium is explored. The system consists of a binary mixture of a relatively smaller sized solute, whose size is varied and a larger sized solvent interacting via Lennard-Jones potential. Calculations have been performed at three different reduced densities of 0.7, 0.8, and 0.933. These simulations show that diffusivity exhibits a maximum for some intermediate size of the solute when the solute diameter is varied. The maximum is found at the same size of the solute at all densities which is at variance with the prediction of the levitation effect. In order to understand this anomaly, additional simulations were carried out in which the degree of disorder has been varied while keeping the density constant. The results show that the diffusivity maximum gradually disappears with increase in disorder. Disorder has been characterized by means of the minimal spanning tree. Simulations have also been carried out in which the degree of disorder is constant and only the density is altered. The results from these simulations show that the maximum in diffusivity now shifts to larger distances with decrease in density. This is in agreement with the changes in void and neck distribution with density of the host medium. These results are in excellent agreement with the predictions of the levitation effect. They suggest that the effect of disorder is to shift the maximum in diffusivity towards smaller solute radius while that of the decrease in density is to shift it towards larger solute radius. Thus, in real systems where the degree of disorder is lower at higher density and vice versa, the effect due to density and disorder have opposing influences. These are confirmed by the changes seen in the velocity autocorrelation function, self part of the intermediate scattering function and activation energy. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.3701619]

Effect of pressure on the ionic conductivity of Li+ and Cl- ions in water

A molecular dynamics simulation study of aqueous solution of LiCl is reported as a function of pressure. Experimental measurements of conductivity of Li+ ion as a function of pressure shows an increase in conductivity with pressure. Our simulations are able to reproduce the observed trend in conductivity. A number of relevant properties have been computed in order to understand the reasons for the increase in conductivity with pressure. These include radial distribution function, void and neck distributions, hydration or coordination numbers, diffusivity, velocity autocorrelation functions, angles between ion-oxygen and dipole of water as well as OH vector, mean residence time for water in the hydration shell, etc. These show that the increase in pressure acts as a structure breaker. The decay of the self part of the intermediate scattering function at small wave number k shows a bi-exponential decay at 1 bar which changes to single exponential decay at higher pressures. The k dependence of the ratio of the self part of the full width at half maximum of the dynamic structure factor to 2Dk(2) exhibits trends which suggest that the void structure of water is playing a role. These support the view that the changes in void and neck distributions in water can account for changes in conductivity or diffusivity of Li+ with pressure. These results can be understood in terms of the levitation effect. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.4756909]

Relation Between the Diffusivity, Viscosity, and Ionic Radius of LiCl in Water, Methanol, and Ethylene Glycol: A Molecular Dynamics Simulation

A molecular dynamics (MD) investigation of LiCl in water, methanol, and ethylene glycol (EG) at 298 K is reported. Several; structural and dynamical properties of the ions as well as the solvent such as self-diffusivity, radial distribution functions, void and neck distributions, velocity autocorrelation functions, and mean residence times of solvent in the first solvation shell have been computed. The results show that the reciprocal relationship between the self-diffusivity of the ions and the viscosity is valid in almost all solvents with the exception of water. From an analysis of radial distribution functions and coordination numbers the nature of hydrogen bonding within the solvent and its influence on the void and neck distribution becomes evident. It is seen that the solvent solvent interaction is important in EG while solute solvent interactions dominate in water and methanol. From Voronoi tessellation, it is seen that the voids and necks within methanol are larger as compared to those within water or EG. On the basis of the void and neck distributions obtained from MD simulations and literature experimental data of limiting ion conductivity for various ions of different sizes we show that there is a relation between the void and neck radius on e one hand and dependence of conductivity on the ionic radius on the other. It is shown that the presence of large diameter voids and necks in methanol is responsible for maximum in limiting ion conductivity (lambda(0)) of TMA(+), while in water in EG, the maximum is seen for Rb+. In the case of monovalent anions, maximum in lambda(0) as a function ionic radius is seen for Br- in water EG but for the larger ClO4- ion in methanol. The relation between the void and neck distribution and the variation in lambda(0) with ionic radius arises via the Levitation effect which is discussed. These studies show the importance of the solvent structure and the associated void structure.

Analytical calculations of Eulerian and Lagrangian time correlations in turbulent shear flows

The Taylor series expansion method is used to analytically calculate the Eulerian and Lagrangian time correlations in turbulent shear flows. The short-time behaviors of those correlation functions can be obtained from the series expansions. Especially, the propagation velocity and sweeping velocity in the elliptic model of space-time correlation are analytically calculated and further simplified using the sweeping hypothesis and straining hypothesis. These two characteristic velocities mainly determine the space-time correlations.