Velocity Autocorrelation Function Research Articles

Time Fractional Diffusion Equations and Analytical Solvable Models

The anomalous diffusion of a particle that moves in complex environments is analytically studied by means of the time fractional diffusion equation. The influence on the dynamics of a random moving particle caused by a uniform external field is taken into account. We extract analytical solutions in terms either of the Mittag-Leffler functions or of the M- Wright function for the probability distribution, for the velocity autocorrelation function as well as for the mean and the mean square displacement. Discussion of the applicability of the model to real systems is made in order to provide new insight of the medium from the analysis of the motion of a particle embedded in it.

Dynamics of a model colloidal suspension from dilute to freezing.

Molecular dynamics simulation was used to study a model colloidal suspension at a range of packing fractions from the dilute limit up to the freezing point. This study builds on previous work by the authors which modeled the colloidal particles with a hard core surrounded by a Weeks-Chandler-Anderson potential with modified interaction parameters, and included an explicit solvent. In this work, we study dynamical properties of the model by first calculating the velocity autocorrelation function, the self-diffusion coefficient, and the mutual diffusion coefficient. We also perform detailed calculations of the colloidal particle intermediate scattering function to study the change in dynamics leading up to the freezing point, and to determine whether the current model can be used to interpret light scattering experiments. We then perform a multiexponential analysis on the intermediate scattering function results and find that the data are fitted well by the sum of two exponentials, which is in line with previous analysis of experimental colloidal suspensions. The amplitudes and decay coefficients of the two modes are determined over a large range of wave vectors at packing fractions leading up to the freezing point. We found that the maximum wave vector at which macroscopic diffusive behavior was observed decreased as the packing fraction increased, and a simple extrapolation shows the maximum wave vector going to zero at the melting point. Lastly, the ratio of the two decay coefficients is compared to the scaling law proposed by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.771]. It was found that the ratio was not constant, but instead was wave vector dependent.

Asymptotic neutron scattering laws for anomalously diffusing quantum particles.

The paper deals with a model-free approach to the analysis of quasielastic neutron scattering intensities from anomalously diffusing quantum particles. All quantities are inferred from the asymptotic form of their time-dependent mean square displacements which grow ∝t(α), with 0 ≤ α < 2. Confined diffusion (α = 0) is here explicitly included. We discuss in particular the intermediate scattering function for long times and the Fourier spectrum of the velocity autocorrelation function for small frequencies. Quantum effects enter in both cases through the general symmetry properties of quantum time correlation functions. It is shown that the fractional diffusion constant can be expressed by a Green-Kubo type relation involving the real part of the velocity autocorrelation function. The theory is exact in the diffusive regime and at moderate momentum transfers.

DESENVOLVIMENTO DE UM MODELO MATEMÁTICO PARA DESCREVER A DISPERSÃO DE CONTAMINANTES EM SITUAÇÕES DE MEANDRO DO VENTO HORIZONTAL

In this work it’s derived a mathematical model to describe the meandering contaminants dispersion in the planetary boundary layer. The model employs a heuristic functional form for the velocity autocorrelation function which describes the negatives lobes observed in meandering data. Furthermore, the model is derived from the linearization of the Langevin equation. The mathematical equations, that represent the longitudinal and lateral wind components describe the bi-dimensional pattern of the meandering flow and can be utilized to calculate the contaminant concentrations occurring in situations in which the meandering phenomenon plays a decisive role.

Atomistic Insight into Tetraalkylphosphonium Bis(oxalato)borate Ionic Liquid/Water Mixtures. 2. Volumetric and Dynamic Properties.

Atomistic molecular dynamics simulations have been performed to investigate volumetric quantities and dynamic properties of binary trihexyltetradecylphosphonium bis(oxalato)borate ([P6,6,6,14][BOB]) ionic liquid (IL)/water mixtures with different water concentrations. The predicted liquid densities for typical [P6,6,6,14][BOB] IL/water mixtures are consistent with available experimental data with a relative discrepancy of less than 3%. The liquid densities and excess molar volumes of all studied [P6,6,6,14][BOB] IL/water mixtures are characterized by concave and convex features, respectively, within full water concentration range. The dynamic properties of [P6,6,6,14] cations, [BOB] anions, and water molecules are particularly analyzed through calculation of velocity autocorrelation functions, diffusion coefficients, and reorientational autocorrelation functions and correlation times. The translational and reorientational mobilities of three species become faster upon increasing water concentration in [P6,6,6,14][BOB] IL/water mixtures and present complex dynamical characteristics arising from three distinct microscopic diffusion features within the full water concentration range. The obtained striking volumetric quantities and particular dynamic properties are well correlated to microscopic liquid structural organization and distinct local ionic environment of all studied [P6,6,6,14][BOB] IL/water mixtures.

Static and dynamic properties of supercooled water in small nanotubes.

The static and dynamic properties of water in small silicon-carbide and carbon nanotubes have been studied over the temperature range 100 K-298 K, using extensive molecular dynamics simulations. The computed properties include the radial distribution function, the cage correlation function, the space-time autocorrelation function, the velocity autocorrelation function, and the self-diffusivity. They all indicate that, under the conditions that we study, water does not freeze in small nanotubes; the Stokes-Einstein relation breaks down, and the self-diffusivity exhibits a transition around 230 K, very close to 228 K, the temperature at which a fragile-to-strong dynamic crossover is supposed to happen. The cage correlation function C(t) decays according to a stretched-exponential function, C(t) ∼ exp[ - (t/τ)(β)], where τ is a relaxation time and β is a topological exponent.

Rare events out of equilibrium: mobility through the liquid–liquid interface

Transition state theory provides a well established means to compute the rate at which rare events occur; however, this is strictly an equilibrium approach. Here we consider a nonequilibrium problem of this nature in the form of transport through a liquid–liquid interface. When two immiscible liquids are coexisting in equilibrium, there will be a certain amount of mixing between the two phases, resulting in a finite linear mobility across the liquid–liquid interface. We derive an exact relationship between the mobility and the local diffusion in the direction perpendicular to the interface. We compute the mobility using both nonequilibrium molecular dynamics and a variety of linear response type approaches, with accurate agreement being obtained for the best of these. Our analysis makes it clear how the local diffusion is influenced by the inhomogeneities of the interface, even when at a distance from it. This nonlocal character to the mobility has not been appreciated before and results in a strong variation in the local diffusion, which is formally coupled to the variation in the potential of mean force. The nonlocal aspect of the diffusion requires the velocity autocorrelation function to be integrated out to far longer times than is the case for homogeneous liquids, and requires special care with regard to the choice of numerical approach.

Statistical properties of the surface velocity field in the northern Gulf of Mexico sampled by GLAD drifters

AbstractThe Grand LAgrangian Deployment (GLAD) used multiscale sampling and GPS technology to observe time series of drifter positions with initial drifter separation of O(100 m) to O(10 km), and nominal 5 min sampling, during the summer and fall of 2012 in the northern Gulf of Mexico. Histograms of the velocity field and its statistical parameters are non‐Gaussian; most are multimodal. The dominant periods for the surface velocity field are 1–2 days due to inertial oscillations, tides, and the sea breeze; 5–6 days due to wind forcing and submesoscale eddies; 9–10 days and two weeks or longer periods due to wind forcing and mesoscale variability, including the period of eddy rotation. The temporal e‐folding scales of a fitted drifter velocity autocorrelation function are bimodal with time scales, 0.25–0.50 days and 0.9–1.4 days, and are the same order as the temporal e‐folding scales of observed winds from nearby moored National Data Buoy Center stations. The Lagrangian integral time scales increase from coastal values of 8 h to offshore values of approximately 2 days with peak values of 3–4 days. The velocity variance is large, , the surface velocity statistics are more anisotropic, and increased dispersion is observed at flow bifurcations. Horizontal diffusivity estimates are in coastal regions with weaker flow to in flow bifurcations, a strong jet, and during the passage of Hurricane Isaac. The Gulf of Mexico surface velocity statistics sampled by the GLAD drifters are a strong function of the feature sampled, topography, and wind forcing.

Lattice Dynamics and Thermal Transport in Superionic Conducting Li7La3Zr2O12 Garnet

Highly conductive solid electrolytes (SE) that are stable in contact with Li and cathodes are required for all solid state battery or high energy Li-S and Li-air batteries. With the advantages of high conductivity (>10-4 S/cm) and excellent stability, Li-stuffed garnet Li7La3Zr2O12 has emerged as one of most promising SEs for Li batteries. Even though the Li ionic dynamics in garnet have been studied extensively, the couplings between Li transport and the vibrations of framework atoms, namely phonons, have seldom been studied, and the controversies in Li transport mechanism and superionic transition have not been settled. In this study, we synthesized phase-pure tetragonal and cubic Li7La3Zr2O12 using a solid state reaction method, and systematically studied their thermal conduction, lattice dynamics, and phase transition. Cubic phase, stabilized by 0.3 mol aluminum, shows much lower thermal conductivity k than that of tetragonal phase, e. g. 1.7 vs. 4.3 W/m-K at 70 K and 1.4 vs. 2.4 W/m-K at 300 K. More specifically, k of cubic phase is almost temperature independent and approaches the amorphous limit, indicating very strong phonon scattering; while the tetragonal phase shows a decreased k with increasing temperature, typical of crystalline solids with the Umklapp processes dominating phonon scatterings. By examining the lattice dynamics and velocity auto–correlation function (VAF) in the cubic phase, we found a high frequency hopping mechanism of Li rather than a liquid-like diffusion mechanism, and Li atoms predominantly contribute to the low energy lattice excitations. This indicates that Li atoms ‘rattle’ intensively in their equilibrium position before hopping to another sites, and the ‘rattling-like’ mode strongly correlate with lattice phonons to block their transport. Moreover, the Li-framework couplings also cause the instability of Li atoms in their positions and promote the hopping responsible for the high ionic conductivity. Our study provides additional understanding of ionic transport mechanism in garnet and insights into thermal management of all solid state batteries.

Random walks with thermalizing collisions in bounded regions: Physical applications valid from the ballistic to diffusive regimes

The behavior of a spin undergoing Larmor precession in the presence of fluctuating fields is of interest to workers in many fields. The fluctuating fields cause frequency shifts and relaxation which are related to their power spectrum, which can be determined by taking the Fourier transform of the auto-correlation functions of the field fluctuations. Recently we have shown how to calculate these correlation functions for all values of mean free path (ballistic to diffusive motion) in finite bounded regions, using the model of persistent continuous time random walks (CTRW) for particles subject to scattering by fixed (frozen) scattering centers so that the speed of the moving particles is not changed by the collisions. In this work we show how scattering with energy exchange from an ensemble of scatterers in thermal equilibrium can be incorporated into the CTRW. We present results for 1,2 and 3 dimensions. The results agree for all these cases contrary to the previously studied 'frozen' models. Our results for the velocity autocorrelation function show a long time tail $\left( \sim t^{-1/2}\right) $, which we also obtain from conventional diffusion theory, with the same power, independent of dimensionality. Our results are valid for any Markovian scattering kernel as well as any kernel based on a scattering cross section $\sim1/v.$

Experimental Measurement of Self-Diffusion in a Strongly Coupled Plasma

We present a study of the collisional relaxation of ion velocities in a strongly coupled, ultracold neutral plasma on short timescales compared to the inverse collision rate. Non-exponential decay towards equilibrium for the average velocity of a tagged population of ions heralds non-Markovian dynamics and a breakdown of assumptions underlying standard kinetic theory. We prove the equivalence of the average-velocity curve to the velocity autocorrelation function, a fundamental statistical quantity that provides access to equilibrium transport coefficients and aspects of individual particle trajectories in a regime where experimental measurements have been lacking. From our data, we calculate the ion self-diffusion constant. This demonstrates the utility of ultracold neutral plasmas for isolating the effects of strong coupling on collisional processes, which is of interest for dense laboratory and astrophysical plasmas.

Complete Assignment of the Infrared Spectrum of the Gas-Phase Protonated Ammonia Dimer.

The infrared (IR) spectrum of the ammoniated ammonium dimer is more complex than those of the larger protonated ammonia clusters due to close-lying fundamental and combination bands and possible Fermi resonances (FR). To date, the only theoretical analysis involved partial dimensionality quantum nuclear dynamic simulations, assuming a symmetric structure (D3d) with the proton midway between the two nitrogen atoms. Here we report an extensive study of the less symmetric (C3v) dimer, utilizing both second order vibrational perturbation theory (VPT2) and ab initio molecular dynamics (AIMD), from which we calculated the Fourier transform (FT) of the dipole-moment autocorrelation function (DACF). The resultant IR spectrum was assigned using FTed velocity autocorrelation functions (VACFs) of several interatomic distances and angles. At 50 K, we have been able to assign all 21 AIMD fundamentals, in reasonable agreement with MP2-based VPT2, about 30 AIMD combination bands, and a difference band. The combinations involve a wag or the NN stretch as one of the components, and appear to follow symmetry selection rules. On this basis, we suggest possible assignments of the experimental spectrum. The VACF-analysis revealed two possible FR bands, one of which is the strongest peak in the computed spectrum. Raising the temperature to 180 K eliminated the "proton transfer mode" (PTM) fundamental, and reduced the number of observed combination bands and FRs. With increasing temperature, fundamentals red-shift, and the doubly degenerate wags exhibit larger anharmonic splittings in their VACF bending spectra. We have repeated the analysis for the H3ND(+)NH3 isotopologue, finding that it has a simplified spectrum, with all the strong peaks being fundamentals. Experimental study of this isotopologue may thus provide a good starting point for disentangling the N2H7(+) spectrum.

Properties of solvate shells and the mobility of ions, according to molecular dynamics data

The solvate shells of an ion, its velocity autocorrelation function, and diffusion coefficient D are found, and the interrelations between them are analyzed. A single ion in the system of atoms of a liquid is considered a model system. The interaction between the ion and atoms of the liquid is described by polarization potential U(r); the interaction between atoms of the liquid alone is described by the Lennard–Jones potential. A classical molecular dynamics method is used. Five solvate shells around the ion are found, and the lifetimes of atoms on each shell are calculated. It is found that the velocity autocorrelation function is of a vibrating nature. The spectrum of the autocorrelator and the frequency of cluster vibrations in a linear approximation are compared. Dependences D on parameters of potential U(r) are found. No dependence D on the ion mass is found; this is explained by solvation. The Einstein–Stokes formula and the HSK approximation are used in discussing the results. It is shown that at small radii of the ion, dependence D on parameters U(r) is described by such a model. When the ion radius is increased, the deviation from this dependence and an increase in D are observed. The results are compared to experimental mobilities of O 2 - and Ar 2 + ions in liquid argon.

Alternative first-principles calculation of entropy for liquids.

We present an alternative method for interpreting the velocity autocorrelation function (VACF) of a fluid with application to extracting the entropy in a manner similar to the methods developed by Lin et al. [J. Chem. Phys. 119, 11792 (2003)]JCPSA60021-960610.1063/1.1624057 and improved upon by Desjarlais [Phys. Rev. E 88, 062145 (2013)]PLEEE81539-375510.1103/PhysRevE.88.062145. The liquid VACF is decomposed into two components, one gas and one solid, and each contribution's entropic portion is calculated. However, we fit both the gas and solid portions of the VACF in the time domain. This approach is applied to a single-component liquid (a two-phase model of liquid Al at the melt line) and two different two-component systems: a superionic-to-superionic (bcc to fcc) phase transition in H_{2}O at high temperatures and pressures and a metastable liquid state of MgO. For all three examples, comparisons to existing results in the literature demonstrate the validity of our alternative.

Ionic transport in high-energy-density matter.

Ionic transport coefficients for dense plasmas have been numerically computed using an effective Boltzmann approach. We have developed a simplified effective potential approach that yields accurate fits for all of the relevant cross sections and collision integrals. Our results have been validated with molecular-dynamics simulations for self-diffusion, interdiffusion, viscosity, and thermal conductivity. Molecular dynamics has also been used to examine the underlying assumptions of the Boltzmann approach through a categorization of behaviors of the velocity autocorrelation function in the Yukawa phase diagram. Using a velocity-dependent screening model, we examine the role of dynamical screening in transport. Implications of these results for Coulomb logarithm approaches are discussed.

Molecular dynamics approach to the structural characterization and transport properties of poly(acrylonitrile)/N,N-dimethylformamide solutions

Poly(acrylonitrile) (PAN) in N,N-dimethylformamide (DMF) is a popular solution for producing large variety of polymer products. To precisely describe the behaviours of PAN and DMF in the synthesis processes, it is significant to call for more details about the structure, some thermodynamic and dynamical properties of PAN-DMF solutions. A PAN-DMF solution was simulated via molecular dynamics with an all-atom OPLS type potential in both the NPT and NVT ensembles. The simulation results were evaluated with quantum mechanical calculations (MP2/6-311++G(d,p) and counterpoise procedure) and were compared with available experimental results. The liquid structure was illustrated with pair correlation functions and transport and dynamics properties were calculated with the mean-square displacements MSD and the velocity autocorrelation functions. The strong H-bonds of CN⋯HCO, CH⋯OCH and CH2⋯OCH, with distances of 2.55Å, 2.55Å and 2.65Å, respectively, were found. The largest interaction energy of −7.157kcal/mol between DMF molecules and PAN molecules was found at 4.9Å center-of-mass distance. A potential profile of intermolecular interaction of DMF with PAN along the interaction distance was presented, clearly showing an increase of DMF vaporisation heat when it getting close to PAN molecules. This provided very useful information to analyse the vaporisation behaviours of DMF at the microscopic level, which is essential to comprehensively understand molecular rearrangements towards the design of synthetic processes. The impact of the presence of the PAN on the DMF solution properties were also benchmarked with pure DMF solution.

From local to hydrodynamic friction in Brownian motion: A multiparticle collision dynamics simulation study.

The friction and diffusion coefficients of rigid spherical colloidal particles dissolved in a fluid are determined from velocity and force autocorrelation functions by mesoscale hydrodynamic simulations. Colloids with both slip and no-slip boundary conditions are considered, which are embedded in fluids modeled by multiparticle collision dynamics with and without angular momentum conservation. For no-slip boundary conditions, hydrodynamics yields the well-known Stokes law, while for slip boundary conditions the lack of angular momentum conservation leads to a reduction of the hydrodynamic friction coefficient compared to the classical result. The colloid diffusion coefficient is determined by integration of the velocity autocorrelation function, where the numerical result at shorter times is combined with the theoretical hydrodynamic expression for longer times. The suitability of this approach is confirmed by simulations of sedimenting colloids. In general, we find only minor deviations from the Stokes-Einstein relation, which even disappear for larger colloids. Importantly, for colloids with slip boundary conditions, our simulation results contradict the frequently assumed additivity of local and hydrodynamic diffusion coefficients.

Effect of Dipole‐Dipole Interaction on the Compressional Oscillations in Two‐Dimensional Yukawa liquids

AbstractThe paper presents the study of oscillations in the two‐dimensional Yukawa liquids where the interparticle interaction potential in addition to the Yukawa term has the dipole‐dipole interaction term. The frequency of longitudinal oscillations was studied using the Fourier transform of the velocity autocorrelation function. It was found that in the two‐dimensional Yukawa liquids the longitudinal oscillation frequency of particles becomes sensitive to variations in the coupling parameter if even weak additional dipole‐dipole interaction between particles exists. (© 2016 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Sum Rule Constraints and the Quality of Approximate Kubo-Transformed Correlation Functions.

In this work, a general protocol for evaluating the quality of approximate Kubo correlation functions of nontrivial systems in many dimensions is discussed. We first note that the generalized deconvolution of the Kubo transformed correlation function onto a time correlation function at a given value τ in imaginary time, such that 0 < τ < βℏ, leads to a series of sum rules applicable to the nth derivative of the Kubo function and whose iterative extension allows us to link derivatives of different order in the corresponding correlation functions. We focus on the case when τ = βℏ/2, for which all deconvolution kernels become real valued functions and their asymptotic behavior at long times exhibits a polynomial divergence. It is then shown that thermally symmetrized static averages, and the averages of the corresponding time derivatives, are ideally suited to investigate the quality of approximate Kubo correlation functions at successively larger (and up to arbitrarily long) times. This overall strategy is illustrated analytically for a harmonic system and numerically for a multidimensional double-well potential and a Lennard-Jones fluid. The analysis includes an assessment of RPMD position autocorrelation results as a function of the number of dimensions in a double-well potential and of the RPMD velocity autocorrelation function of liquid neon at 30 K.

Calculating Transmembrane Diffusivity

The rate of unfacilitated diffusion of solutes through a lipid bilayer is an important factor in cell signaling, toxicology, and pharmacokinetics. The solubility-diffusion model allows the permeability coefficient of a solute to be calculated using molecular dynamics simulations [1]. This method requires only the potential of mean force and diffusion coefficient of the solute along the transmembrane axis. Accurate calculation of the transmembrane diffusion coefficient profile has proven difficult because the diffusivity depends strongly on the position of the solute in the membrane and converges slowly. Hummer proposed a simple method to calculate position-dependent diffusivity from the position-autocorrelation function of a restrained molecular dynamics simulations, but the values calculated in the lipid tail region can have large uncertainties and are sensitive to the method of integration, etc. We show that long-time correlations of the solute within the membrane are the cause of these issues. An alternative method based on the Laplace transform of the velocity autocorrelation function avoids this issue and provides a straightforward and reliable method of calculating transmembrane diffusion coefficient profiles. This method will facilitate calculation of the rates of unfacillitated membrane permeation. Our implementation of this method is available for free online [5].[1] Marrink, Berendsen, J. Phys. Chem., 1994, 98, 4155, http://dx.doi.org/10.1021/j100066a040.[2] Riahi, Rowley, J. Am. Chem. Soc., 2014, 136, 15111 http://dx.doi.org/10.1021/ja508063s.[3] Hummer, New J. Phys. 2005, 7, 34, http://dx.doi.org/10.1088/1367-2630/7/1/034.[4] Woolf and Roux. J. Am. Chem. Soc. 1994, 116, 5916, http://dx.doi.org/10.1021/ja00092a048.[5] GitHub: https://github.com/RowleyGroup/ACFCalculator.