Hard-sphere Molecular Dynamics Research Articles

Examination of Couette Flow with a Pressure Gradient and Heat Conduction Using Molecular Dynamics Simulation

As computer capabilities improve, Molecular Dynamics simulations are becoming more important for solving various flow problems. In this study, Couette and Poiseuille flows at different wall temperatures were investigated using a hard-sphere Molecular Dynamics simulation approach. Although a low spacing ratio was used in the simulations, the results are valid for rarefied gas flows when proper scaling based on the Knudsen number was used because only binary collisions with a hard-sphere model were considered. The main focus of this study was the examination of the effects of various wall speeds, pressure gradients, and wall temperatures. A pressure gradient was generated by developing a modified selective periodicity condition in the flow direction. With the combined effect of the pressure gradient and the wall velocities, subsonic, transonic, and supersonic speeds in nanochannels were examined. With the combination of different parameters, 1260 simulation cases were conducted. The results showed that there are temperature and velocity slips that are dependent on not only the temperature and velocity values but also on the magnitudes of a pressure gradient. The pressure gradient also caused nonlinearities in temperature and velocity profiles.

The divergence of nearby trajectories in soft-sphere DEM

The n-body instability is investigated with the soft-sphere discrete element method. The divergence of nearby trajectories is quantified by the dynamical memory time. Using the inverse proportionality between the dynamical memory time and the largest Lyapunov exponent, the soft-sphere discrete element method results are compared to previous hard-sphere molecular dynamics data for the first time. Good agreement is observed at low concentrations and the degree of instability is shown to increase asymptotically with increasing spring stiffness. At particle concentrations above 30%, the soft-sphere Lyapunov exponents increase faster than the corresponding hard-sphere data. This paper concludes with a demonstration of how this case study may be used in conjunction with regression testing and code verification activities.

Particle sedimentation using hybrid Lattice Boltzmann-immersed boundary method scheme

Particle collisions play a critical role in determining the fluid-particle interaction. Thus, particulate matters and their interaction within a flow need an accurate treatment through felicitous composite methods. Accordingly, we present a numerical simulation based on a combined Lattice Boltzmann Method (LBM), Immersed Boundary Method (IBM), and Hard Sphere Molecular Dynamics (HSMD) scheme, with applications. This simulation method aims to understand spherical point-particle sedimentation in an incompressible Newtonian fluid. A Stokes' drag model implements the hydrodynamic interaction using Ladd's LBM scheme and a momentum exchange based IBM. The HSMD evaluates the discrete particles' kinematics and trajectory. We simulate a 3D single particle settling case as a benchmark. The simulation result exhibits a close match between the terminal particle velocity and an analytical result; it reproduces the expected exponential behavior. Finally, a sedimentation case involving 7200 particles demonstrates the composite schemes' capability to solve the complex interactions. This article discusses the interesting observations and results.

Scaling the transport properties of molecular and ionic liquids

There are many forms of scaling for the transport properties of liquids and liquid solutions. This review concentrates on the application of models based on (a) hard-sphere molecular dynamics results for dense molecular fluids, (b) Stokes-Einstein-Sutherland (SES) scaling of experimental transport properties with the viscosity for both molecular and ionic liquids, and (c) thermodynamic or density scaling, again for both types. An alternative, but similar, approach to SES scaling for ionic liquids employs so-called “distinct diffusion coefficients” incorporating both conductivity and self-diffusion terms in Walden-type plots. In ionic liquid families, related compounds are found to scale on to common curves in some cases. Recently thermodynamic scaling, using (TVγ) as the scaling function has been introduced. For molecular liquids, γ can be related to the exponent of the repulsive part of the potential, but its meaning is less clear for ionic liquids that consist of large charged ions and have both significant van der Waals and Coulombic interactions. The further development of scaling offers promise for the future in relating the behaviour of ionic liquids to those of other liquid classes.

Viscosity of a neutron gas

Abstract The viscosity of nucleonic matter and in particular a neutron gas is studied both classically and in a quantum mechanical description. The collisions between particles are modeled as hard sphere scattering as a baseline for comparison and as scattering from an attractive square well potential. Properties associated with the unitary limit are developed which are shown to be approximately realized for a system of neutrons. The issue of near perfect fluid behavior of neutron matter is remarked on. Using some results from hard sphere molecular dynamics studies near perfect fluid behavior is discussed further.

Time dependence and density inversion in simulations of vertically oscillated granular layers

We study a layer of grains atop a plate which oscillates sinusoidally in the direction of gravity, using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order as well as hard-sphere molecular dynamics simulations. For high accelerational amplitudes of the plate, the layer exhibits a steady-state "density inversion" in which a high-density portion of the layer is supported by a lower-density portion. At low accelerational amplitudes, the layer exhibits oscillatory time dependence that is strongly correlated to the motion of the plate. We show that continuum simulations yield results consistent with molecular dynamics results in both regimes.

Molecular dynamics simulation of compressible hot/cold moving lid-driven microcavity flow

Hard-sphere molecular dynamics simulations of lid-driven microcavity gas flow with various subsonic speeds and lid temperatures are conducted. Simulations with faster and colder lids show streamlines of stronger primary vortices. Variations of mass and energy centers with respect to lid speed and temperature are examined. Center of energy is less sensitive to employed lid conditions than center of gravity is. Although moving lid imparts energy into fluid, due to change of impingement rates on the walls of fixed temperature, average energy within the cavity seems quite insensitive to the subsonic lid speed. Behavior of compressibility at both top corners is observed even at low Mach numbers widely considered within incompressible flow region. While high Knudsen number causes considerable property slips near the lid, two-dimensional pressure, density, and temperature plots of excellent quality are generated. Results are promising in use of molecular dynamics simulations for compressible vortex flow analyses while providing insights for understanding microfluidics and nanofluidics in context of molecular mass, momentum and heat transfer in microscale and nanoscale systems.

Anisotropic mean free path in simulations of fluids traveling down a density gradient

Hard sphere molecular dynamics simulations are used to study the mean free path of molecules traveling down a density gradient at fluid densities ranging between 0.05sigma(-3) and 0.7sigma(-3). Gradients are developed using semipermeable boundaries in the x-direction and, as a result, a net flow develops in the positive x-direction. Over the course of the simulation, the free paths of colliding molecules are calculated and it was determined that the mean free path in the positive x-direction is greater than the mean free path in the negative x-direction at each density studied. These results are compared to the mean free paths in the positive and negative y- and z-directions (in which there is no net flow) and the distribution of free paths for molecules traveling in the positive and negative x-directions gives insight into the physics of the system. In addition, the dependency of the mean free path on speed is studied and compared to kinetic theory predictions. The results have application in the modification of the classical model of diffusion for low density systems undergoing flow in which the mean free path is finite, large, and can be anisotropic.

Thermodynamic pressures for hard spheres and closed-virial equation-of-state

Hard-sphere molecular dynamics (MD) simulation results, with six-figure accuracy in the thermodynamic equilibrium pressure, are reported and used to test a closed-virial equation-of-state. This latest equation, with no adjustable parameters except known virial coefficients, is comparable in accuracy both to Padé approximants, and to numerical parameterizations of MD data. There is no evidence of nonconvergence at stable fluid densities. The virial pressure begins to deviate significantly from the thermodynamic fluid pressure at or near the freezing density, suggesting that the passage from stable fluid to metastable fluid is associated with a higher-order phase transition; an observation consistent with some previous experimental results. Revised parameters for the crystal equation-of-state [R. J. Speedy, J. Phys.: Condens. Matter 10, 4387 (1998)] are also reported.

Stochastic Hard-Sphere Dynamics for Hydrodynamics of Nonideal Fluids

A novel stochastic fluid model is proposed with a nonideal structure factor consistent with compressibility, and adjustable transport coefficients. This stochastic hard-sphere dynamics (SHSD) algorithm is a modification of the direct simulation Monte Carlo algorithm and has several computational advantages over event-driven hard-sphere molecular dynamics. Surprisingly, SHSD results in an equation of state and a pair correlation function identical to that of a deterministic Hamiltonian system of penetrable spheres interacting with linear core pair potentials. The fluctuating hydrodynamic behavior of the SHSD fluid is verified for the Brownian motion of a nanoparticle suspended in a compressible solvent.

Fluctuation theorem for a single particle in a moving billiard: Experiments and simulations

A single particle within a periodically driven Sinai-billiard-like system is tracked experimentally and is simulated via hard-sphere molecular dynamics. Both experiments and simulations confirm the fluctuation theorem (FT); thus, in this setup, one driven particle is sufficient for the FT to hold. Furthermore, in the simulations the chaoticity of the system can be adjusted using the restitution coefficient of collisions. It is also shown that the FT breaks down when unstable periodic orbits appear for small restitution coefficients.

Osmosis in semi-permeable pores: an examination of the basic flow equations based on an experimental and molecular dynamics study

Classically ‘semi-permeable’ pores are generally considered to mediate osmotic flow at a rate dependent upon the hydraulic conductance of the pore and the difference in water potential. The shape or size of the solute molecules is not considered to exert a first-order effect on the flow rate nor is the hydraulic conductance thought to be solute dependent. By the experimental measurement of osmosis in the biological pore AQP (aquaporin) and hard-sphere molecular dynamics simulation of a model pore, we show here that the solute radius can have a profound effect on the osmotic flow rate, causing it to decline steeply with decreasing solute radius.Using a simple non-equilibrium thermodynamic theory, we propose that an additional ‘osmotic flow coefficient’ is required to describe flows in semi-permeable structures such as AQPs, and that the fall in flow rate with radius represents a conversion from hydraulic to diffusive water flow due to increasing penetration of the pore by the solute. The interaction between the pore geometry and the solute size cannot, therefore, be overlooked, although for every solute the system obeys the criterion for semi-permeability required by basic thermodynamics. The osmotic pore theory therefore reveals a novel and potentially rich structure that remains to be explored in full.

Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulation

We examine consequences of the non-Boltzmann nature of probability distributions for one-particle kinetic energy, momentum, and velocity for finite systems of classical hard spheres with constant total energy and nonidentical masses. By comparing two cases, reflecting walls (NVE or microcanonical ensemble) and periodic boundaries (NVEPG or molecular dynamics ensemble), we describe three consequences of the center-of-mass constraint in periodic boundary conditions: the equipartition theorem no longer holds for unequal masses, the ratio of the average relative velocity to the average velocity is increased by a factor of [N/(N-1)]1/2, and the ratio of average collision energy to average kinetic energy is increased by a factor of N/(N-1). Simulations in one, two, and three dimensions confirm the analytic results for arbitrary dimension.

Simulation of binary hard-sphere systems with 1 : 5 and 1 : 10 size ratios

Hard-sphere molecular dynamics simulations were performed on asymmetric binary systems to obtain pair-correlation functions. Particle-diameter ratios of 1 : 5 and 1 : 10 were applied to mimic the colloid/solvent situation, where the large particles were the colloids and the small ones represented the solvent molecules. The packing fractions were chosen systematically between 0.1–0.5 for the large spheres and 0.0–0.5 for the small ones. We focused on two questions: a) To what extent can one approximate the inter-particle structure of the large spheres in binary systems with one-component hard-sphere data. b) How reliable are the theoretical pair-correlation functions of the Percus-Yevick and Rational Function Approximation models. Overlap integrals of pair-correlation function pairs were calculated to answer quantitatively. The results supported quantitatively the limits of the one-component hard-sphere approximation. On contrary, the selected theoretical methods reproduced the simulation results satisfactorily, if the packing fraction of the solvent was not more than 0.2. The theoretically important contact values of the pair-correlation functions were calculated, too.

Convection in vibrated annular granular beds

The response to vibration of a granular bed, consisting of a standard cylindrical geometry but with the addition of a dissipative cylindrical inner wall, has been investigated both experimentally (using positron emission particle tracking) and numerically (using hard sphere molecular dynamics simulation). The packing fraction profiles and granular temperature distributions (in both vertical and horizontal directions) were determined as a function of height and distance from the axis. The two sets of results were in reasonable agreement. The molecular dynamics simulations were used to explore the behavior of the granular bed in the inner wall-outer wall coefficient of restitution phase space. It was observed that one could control the direction of the toroidal convection rolls by manipulating the relative dissipation at the inner and outer walls via the coefficients of restitution, and with several layers of grains it was seen that double convection rolls could also be formed, a result that was subsequently confirmed experimentally.

Osmosis in small pores: a molecular dynamics study of the mechanism of solvent transport

Osmosis through semi–permeable pores is a complex process by which solvent is driven by its free energy gradient towards a solute–rich reservoir. We have studied osmotic flow across a semi–permeable cylindrical pore using hard–sphere molecular dynamics which simulates osmosis in the absence of attractive forces between solute and solvent. In addition, we recorded the rates of pressure–driven solvent flow and the diffusive flow of labelled solvent under concentration gradients. It is apparent that there are differences, which are radius dependent, between viscous and diffusive solvent permeabilities in small pores.The osmotic flow rate is decreased by allowing solute entry into part of the pore, an effect which is not due to solute obstruction. The flow rate is dependent on the structure of the pore, which for asymmetric pores leads, surprisingly, to flow asymmetry or osmotic rectification. In the absence of any possible viscous rectification at these very low flow rates the effect correlates with changes between diffusive and pressure flows created by the presence of solute, an effect which has been predicted from thermodynamic arguments. The geometry of a semi–permeable pore in relation to the solute size is therefore required to predict the osmotic flow rate, a departure from the classical picture.Finally, by extracting transport parameters from simulations with pure solvent, we examine the departure of observed flow rate from that predicted by continuum mechanics, obtaining drag coefficients which we compare with those derived from hydrodynamics alone.

A Multiple-Heaps Algorithm for Parallel Simulation of Collision Systems

We consider the parallel simulation of collision systems. It has wide application, such as in hard-sphere molecular dynamics simulation for gas dynamics and crystals, as well as in studying molecular collision dynamics of chemical reactions. With detailed analysis, proper data structures are designed so that the central computational task is formulated as a consecutive search for the minimum in the collision time space of O(N2) entries, with multiple updates on O(N) entries in the same space per collision step. The abstraction and formulation enable us to incorporate efficient techniques in computer science into this application, which leads to a heap-based sequential algorithm of O(N log N) time in one typical collision step, where N is the number of particles of the simulated collision system. A parallel algorithm of multiple heaps with a diagonal-oriented mapping is then proposed. We show that the parallel algorithm is load balanced and the parallel time per collision step is O((N/P) log (N2/P)+log P), where P is the number of processors. The parallel algorithm uses two levels of partitioning independently, one in the particle-based physical space and the other in the collision time space. An exchange-shift communication algorithm is presented to bridge the two different partitioning schemes. Besides collision system simulation, the parallel multiple heaps algorithm may find applications in many other computing areas where a heap-based priority queue needs to be maintained, such as in fast level-set methods.

From microscopic simulations to macroscopic material behavior

Abstract One challenge in computational physics is to bridge the gap between microscopic, “atomistic” sizes and the macroscopic length scale of experimental observation. First, an efficient algorithm for hard sphere molecular dynamics is presented, allowing for many-particle simulations of, e.g., granular systems. In the next step, a “micro–macro” transition is introduced which enables continuum quantities like the stress tensor to be accessed. The approach is used for dense and dissipative gases but can also be applied to more complicated systems like membranes.

Phase bubbles and spatiotemporal chaos in granular patterns.

We use inelastic hard sphere molecular dynamics simulations and laboratory experiments to study patterns in vertically oscillated granular layers. The simulations and experiments reveal that phase bubbles spontaneously nucleate in the patterns when the container acceleration amplitude exceeds a critical value, about 7 g, where the pattern is approximately hexagonal, oscillating at one-fourth the driving frequency (f/4). A phase bubble is a localized region that oscillates with a phase opposite (differing by pi) to that of the surrounding pattern; a localized phase shift is often called an arching in studies of two-dimensional systems. The simulations show that the formation of phase bubbles is triggered by undulation at the bottom of the layer on a large length scale compared to the wavelength of the pattern. Once formed, a phase bubble shrinks as if it had a surface tension, and disappears in tens to hundreds of cycles. We find that there is an oscillatory momentum transfer across a kink, and the shrinking is caused by a net collisional momentum inward across the boundary enclosing the bubble. At increasing acceleration amplitudes, the patterns evolve into randomly moving labyrinthian kinks (spatiotemporal chaos). We observe in the simulations that f/3 and f/6 subharmonic patterns emerge as primary instabilities, but that they are unstable to the undulation of the layer. Our experiments confirm the existence of transient f/3 and f/6 patterns.

Algorithms for Particle-Field Simulations with Collisions

We develop an efficient algorithm for detecting collisions among a large number of particles moving in a velocity field, when the field itself is possibly coupled to the particle motions. We build on ideas from molecular dynamics simulations and, as a byproduct, give a literature survey of methods for hard sphere molecular dynamics. We analyze the complexity of the algorithm in detail and present several experimental results on performance which corroborate the analysis. An optimal algorithm for collision detection has cost scaling at least like the total number of collisions detected. We argue, both theoretically and experimentally, that with the appropriate parameter choice and when the number of collisions grows with the number of particles at least as fast as for billiards, the algorithm we recommend is optimal.