Electrostatic interactions in molecular dynamics simulation of a three-dimensional system with periodicity in one direction

The Mellin transform and Poisson summation formula are used to derive an expression for the Coulomb interaction energy of a three-dimensional system with periodicity in one direction. Initially, calculations are performed for interactions characterized by any inverse power and, using the analytical continuation of the energy function, one obtains the final expression for the interaction energy of charges. We consider also a special case when two different charges are located on a line parallel to the periodicity direction. The energy and force expressions are identical to those obtained from the Lekner summation which is simply a sum over reciprocal lattice terms. The convergence behaviour of the Lekner summation is compared with that based on the Ewald type approach.     

Molecular dynamics simulations of liquid water using various long-range electrostatics techniques

Water is one of the most extensively studied molecules, owing to its crucial role in biological processes. The water molecule is both highly polar and highly polarizable. Properties of water computed from molecular simulations are therefore critically dependent on both the intermolecular potential and the method for computing long-range electrostatic corrections. In this paper, the effects of the potential and the long-range electrostatic corrections are quantified for liquid water from 260 to 400?K. Simulations were carried out for a system of 256 molecules in the NVT ensemble. Thermodynamic, structural, dynamical, hydrogen bonding and dielectric properties have been computed for the flexible SPC and rigid SPC, SPC/E, TIP4P, TIP4P-Ew and TIP4P-FQ potentials, using the Lekner, Ewald and reaction field techniques to handle long-range electrostatics. The Lekner method gave the best overall agreement with experimental data, while the reaction field approach produced poorer results. Some measurable differences were found between the Lekner and Ewald techniques. For dielectric properties, the performance of the TIP4P-FQ model was superior relative to other potentials. For 256 molecules, the computational speeds of the Ewald and reaction field methods were found to be 2.5 to 3 times and 3.5 to 5 times faster than the Lekner technique, respectively.     

Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

Yukawa potentials may be long-ranged when the Debye screening length is large. In computer simulations, such long-ranged potentials have to be taken into account with convenient algorithms to avoid systematic bias in the sampling of the phase space. Recently, we provided Ewald sums for quasi-two-dimensional systems with Yukawa interaction potentials [J. Chem. Phys. 126, 056101 (2007); Molec. Phys. paper I of this series]. Sometimes, Lekner sums are used as an alternative to Ewald sums for Coulomb systems. In the present work, we derive the Lekner sums for quasi-two-dimensional systems with Yukawa interaction potentials and we give numerical tests for practical implementations. The main result of this paper is to emphasize that Lekner sums cannot be considered as an alternative to Ewald sums for Yukawa potentials. As a conclusion to this work: Lekner sums should not be used for quasi-two-dimensional systems with Yukawa interaction potentials.     

Effect of electrostatics techniques on the estimation of thermal conductivity via equilibrium molecular dynamics simulation: application to methane hydrate

Equilibrium molecular dynamics (MD) simulations for three system sizes of fully occupied methane hydrate have been performed at around 265 K to estimate the thermal conductivity using the Ewald, Lekner, reaction field, shifted-force and undamped Fennell–Gezelter methods. The TIP4P water model was used in conjunction with a fully atomistic methane potential with which it had been parameterized from quantum simulation. The thermal conductivity was evaluated by integration of the heat flux autocorrelation function (ACF) derived from the Green–Kubo formalism; this approach vas validated by estimation of the average phonon mean free path. The thermal conductivities predicted by non-periodic techniques were in reasonable agreement with the experimental results of 0.62 and 0.68 W/m K, although it was found that the estimates by the non-periodic techniques were up to 25% larger than those of Lekner and Ewald estimates, particularly for larger systems. The results for the Lekner method exhibited the least variation with respect to system size. A decomposition of the heat flux vector into its respective contributions revealed the importance of electrostatic interactions, and how different electrostatic treatments affect the contribution to the thermal conductivity.     

Effects of long-range interactions on the dynamics of ions in aqueous solution

Molecular dynamics simulations have been carried out for a Br⁻ ion in aqueous solution in order to establish the effect of truncation of long-range interactions on the dynamical properties of the ion. Simulations using smooth truncation of the potential at different cutoff radii were carried out and compared to results using the Ewald summation method. It is shown that when small cutoffs are applied (i.e. R_c = 8Å), the calculations yield low ionic diffusion coefficients relative to experiment, as well as short-time dynamical behavior which is inconsistent with the Ewald calculations. As the cutoff is increased, the results approach both the Ewald and the experimental results. In contrast to the results with the truncated potential, the short-time dynamical behavior of the hydrated bromide ion obtained with the Ewald method can be described by simple Langevin dynamics.     

Comparative analysis of interferogram noise filtration using wavelet transform and spin filtering algorithms

The aim of this paper is to analyze 2D fringe pattern denoising performed by two chosen methods based on quasi-1D two-arm spin filter and 2D discrete wavelet transform (DWT) signal decomposition and thresholding. The ultimate aim of this comparison is to estimate which algorithm is better suited for high-accuracy measurements by phase shifting interferometry (PSI) with the phase step evaluation using the lattice site approach. The spin filtering method proposed by Yu et al. (1994) was designed to minimize possible fringe blur and distortion. The 2D DWT also presents such features due to a lossless nature of the signal wavelet decomposition. To compare both methods, a special 2D histogram introduced by Gutman and Weber (1998) is used to evaluate intensity errors introduced by each of the presented algorithms.     

Coulomb interactions in a computer simulation of a system periodic in two directions

The integral representation of the gamma function and the Poisson summation formula are used to calculate the interaction energy of charged particles in a 3-dimensional system periodic in two directions. A parallelogram shape simulation box is considered. Calculations are carried out for interactions described by any inverse power, and analytical continuation of the energy function leads to the final expression for the Coulomb interaction energy. Summation over the simulation box replica along one or the other side of the box base is replaced by summation in reciprocal space. Therefore there are two equivalent formulas for the potential energy that offer the possibility of avoiding slowly convergent series. The energy expressions are identical to those obtained from the Lekner method. The special case is considered where the functions defining the energy are infinite, i.e. when two charges lie on a line parallel to the simulation box side that was chosen to convert real space summation into reciprocal space.     

Calculation of electrostatic fields in ionic crystals based upon the Ewald method

Formulas for the electrostatic potential, field strength and field gradient due to monopoles, dipoles and quadrupoles are given, based upon the Ewald summation method. A convenient choice of the convergence parameter in the Ewald method is demonstrated in some actual calculations. A method to calculate self-consistent electrostatic fields due to self induced dipoles is evaluated. Monopole and self-consistent fields are calculated in Pb₂O₃, βMn₂O₃, FeTiO₃, MnF₃ and Pb₃O₄. The influence of the anion parameter on the electrostatic field is investigated for idealized CdCl₂ and TiO₂. Results of Madelung and induced dipole polarization energy for a series of arbitrary compounds are given. The effect of spontaneous polarization in polar crystals is discussed.     

A behavior model of a magnetorheological fluid in direct shear mode

A model that considers the mutual interaction between particles and the external field in a magnetorheological (MR) fluid is proposed. The model predicts the magnetization, interaction energy density, and shear stress against an imposed strain along the planar direction under a uniform magnetic field. Using the Lekner summation method, slowly convergent magnetization functions are transformed into rapidly convergent ones. By applying the proposed model to an application suitable for commercial clutches and dampers working in direct shear mode, it is revealed that the mean magnetization vector and the associated functional quantities of the MR fluid behave harmonically with respect to the imposed shear strain.     

Behavioral model for magnetorheological fluid under a magnetic field using Lekner summation method

Magnetorheological (MR) fluid is used for various applications due to its controllable viscosity. To predict the behavior of MR fluid under certain three-dimensional (3D) magnetic and shear strain fields, it is essential to model the fluid in an appropriate manner. The behavioral models used in the previous research, however, have serious limitations because most of them oversimplify the inter-particle interactions and employ assumptions valid only under specific geometric configurations and field conditions. In this study, a new model that can predict the behavior of MR fluid under arbitrary 3D magnetic and shear strain fields is proposed. The present work considers an MR fluid configured as a 3D infinite lattice structure. Using the proposed model, the shear stress components themselves, not the dipolar interaction energy, are calculated directly to avoid the mathematical singularity otherwise encountered. The resulting stress functions of the proposed model are transformed into rapidly convergent functions using the Lekner summation method. Finally, the characteristics of the stiffened MR fluid under a magnetic field are investigated using the transformed functions. Numerical computations on the original and transformed functions are performed and compared under selected conditions to ensure the validity and prove the high convergence efficiency of the proposed model.     

Ewald method for polytropic potentials in arbitrary dimensionality

The Ewald summation technique is generalized to power-law 1/| r | ^k potentials in three-, two- and one-dimensional geometries with explicit formulae for all the components of the sums. The cases of short-range, long-range and ‘marginal’ interactions are treated separately. The jellium model, as a particular case of a charge-neutral system, is discussed and the explicit forms of the Ewald sums for such a system are presented. A generalized form of the Ewald sums for a non-cubic (non-square) simulation cell for three- (two-) dimensional geometry is obtained and its possible field of application is discussed. A procedure for the optimization of the involved parameters in actual simulations is developed and an example of its application is presented.     

A simple approach to evaluation of lattice sums

It is shown that starting from a Fourier transform relation one can derive, in a surprisingly simple manner, all the well-known results of lattice summation, that have been obtained so far by a complicated use of the Ewald theta transformation. We show that the Ewald transformation follows directly from the Fourier transform relation.     

Effects of an external electromagnetic field on rutile Tio2: A molecular dynamics study

Non-equilibrium molecular dynamics simulations of the bulk rutile phase of TiO₂ have been carried out in an intense external electromagnetic field with frequency in the microwave to far-infrared range. Simulations were performed in constant-volume ensembles with and without a thermostat coupled to the atomic degrees of freedom, at 298 K and from 298 K to well above the crystal melting temperature, respectively. Fields were applied along the a and c crystallographic directions. Both the Lekner and Ewald techniques were used to handle long-range electrostatics and the impact of different choices of Ewald parameters on the results has been evaluated. It was found that the ions respond rapidly to the field. The peaks and troughs in the Ti-Ti radial distribution functions were sharpened relative to the zero-field case at the instants of maximum electric field intensity, but little evidence of this was found for the Ti-O or O-O distributions. For pure Newtonian dynamics, the 500 GHz field excited the vibrational modes to the greatest extent, raising the system temperature at the fastest rate. For temperatures above 2000-2500 K, the crystal structure was found to melt. In the canonical ensemble, the 50 GHz field led to the greatest enhancement of ionic translational mobility.     

Low energy excitations of a quasi-2D Bose-Einstein condensate

Starting from the Gross-Pitaevskii energy functional of the 3D Bose-Einstein Condensate, we derive approximately the energy functional and the effective coupling constant of the quasi-2D condensate. The evolution of the quasi-2D condensate wave function is studied by a variational method. Low energy excitation spectra for both positive and negative scattering lengths are analyzed. The condition of collapse instability of a quasi-2D Bose gas with attractive particle interaction is also proposed. Received 31 October 2001 / Received in final form 1st March 2002 Published online 28 June 2002     

Error estimates for the fast multipole method

Error estimates for algorithms based on truncations for evaluating electrostatic interactions in molecular dynamics applications are very important for several reasons. For example, the estimates are necessary to establish the validity of the simulations and can be used to estimate various simulation parameters. Very precise estimates have been found for the Ewald method and the related particle mesh Ewald method. However, for the very popular fast multipole method such a precise estimate is not available. In this paper, we illustrate the rather complicated error behavior of the fast multipole method and we use statistical methods to derive an estimate for the root mean square error on the forces. Furthermore, the expected maximum error on the force acting on a single particle is studied. The estimates are tested against errors obtained from simulations and are found to be very precise.     

离心压气机转子内部流场S_1／S_2全三元迭代解

为了比较准三元解和全三元解的差异，验证准三元解在计算离心压气机转子内部流场的准确程度，研究离心压气机转子内部流场全三元流动特性，本文对一有激光测量结果的高压比离心压气机［１］叶轮内部流场进行了全三元迭代计算，分析了Ｓ１／Ｓ２两类流面在叶轮通道内分布形态，比较了两类流面准三元解与全三元解的计算结果，讨论了无粘二次流的分布。并进一步和激光测量值及Ｎ－Ｓ三元直接解进行了详细的比较。     

Quasi-Two-Dimensional Diffusion in Adherent Cells Revealed by Three-Dimensional Single Quantum Dot Tracking

Intracellular diffusion is critical for molecule translocation in cytoplasm and mediates many important cellular processes.Meanwhile,the diffusion dynamics is affected by the heterogeneous cytoplasm.Previous studies on intracellular diffusion are mainly based on two-dimensional(2 D) measurements under the assumption that the three-dimensional(3 D) diffusion is isotropic.However,the real behaviors of 3 D diffusion of molecules in cytoplasm are still unclear.Here,we have built a 3 D single-particle tracking(SPT) microscopy and studied the 3 D diffusion of quantum dots(QDs) in adherent A549 cells.Notably,we found that the intracellular diffusion of QDs is quasi-2 D,with the axial motion being severely confined.Further investigations demonstrated that disrupting the cytoskeleton component or endoplasmic reticulum(ER) does not alter the quasi-2 D diffusion pattern,although ER reduces the diffusion rates and slightly relieves the constraint in the axial diffusion.The preferred quasi-2 D diffusion is quite robust and attributed to the complex cytoarchitectures in the flat adherent cells.With the aid of 3 D SPT method,the quasi-2 D diffusion in cells was revealed,shedding new light on the physical nature of cytoplasm.     

Spectral accuracy in fast Ewald-based methods for particle simulations

A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.