Lees-Edwards Boundary Conditions Research Articles

Implementation of Lees–Edwards periodic boundary conditions for three-dimensional lattice Boltzmann simulation of particle dispersions under shear flow

In this research, the three-dimensional implementation of the Lees-Edwards boundary condition is presented and discussed for the simulation of particulate suspensions. The proposed method is based on the lattice Boltzmann and smoothed profile method for flow and particle motion simulation. This approach eliminates Galilean invariance errors when the particle crosses Lees–Edwards boundaries. Two important issues of particle crossing shear boundaries and corners of a computational domain are discussed in detail. Forces and torques evaluation for a particle that crosses boundaries need special treatment due to the division of the particle into two, four, and eight parts as it traverses one edge, and the intersection of two and three boundaries, respectively. The validation of the present method is performed by using some benchmarks. Furthermore, due to the ability of this algorithm to perform in parallel mode, a CUDA code based on the GPU platform is used to accelerate the computations. The results show that the parallel processing of this method on GPU significantly accelerates the computations.

Molecular Dynamics Investigation of Hyaluronan in Biolubrication.

Aqueous solution of strongly hydrophilic biopolymers is known to exhibit excellent lubrication properties in biological systems, such as the synovial fluid in human joints. Several mechanisms have been proposed on the biolubrication of joints, such as the boundary lubrication and the fluid exudation lubrication. In these models, mechanical properties of synovial fluid containing biopolymers are essential. To examine the role of such biopolymers in lubrication, a series of molecular dynamics simulations with an all-atom classical force field model were conducted for aqueous solutions of hyaluronan (hyaluronic acid, HA) under constant shear. After equilibrating the system, the Lees-Edwards boundary condition was imposed, with which a steady state of uniform shear flow was realized. Comparison of HA systems with hydrocarbon (pentadecane, PD) solutions of similar mass concentration indicates that the viscosity of HA solutions is slightly larger in general than that of PDs, due to the strong hydration of HA molecules. Effects of added electrolyte (NaCl) were also discussed in terms of hydration. These findings suggest the role of HA in biolubirication as a load-supporting component, with its flexible character and strong hydration structure.

Lees–Edwards boundary conditions for translation invariant shear flow: Implementation and transport properties

Molecular dynamics (MD) simulations represent a powerful investigation tool in the field of soft matter. By using shear flows, one can probe the bulk rheology of complex fluids, also beyond the linear response regime, in a way that imitates laboratory experiments. One solution to impose a shear flow in particle-based simulations is the Lees–Edwards technique, which ensures that particles experience shear by imposing rules for motion and interactions across the boundary in the direction of the shear plane. Despite their presentation in 1972, a readily available public implementation of Lees–Edwards boundary conditions has been missing from MD simulation codes. In this article, we present our implementation of the Lees–Edwards technique and discuss the relevant technical choices. We used ESPResSo, the extensible simulation package for research on soft matter, for molecular dynamics simulations which can be used as a reference for other implementers. We illustrate our implementation using bulk dissipative particle dynamics fluids, compare different viscosity measurement techniques, and observe the anomalous diffusion in our samples during continuous and oscillatory shear, in good comparison with theoretical estimates.

Lees-Edwards boundary conditions for the multi-sphere discrete element method

A consistent implementation of Lees-Edwards boundary conditions is proposed for the Multi-Sphere Discrete Element Method, which can mitigate various unphysical effects at the bulk and micro-structural levels. These effects include non-linear velocity profiles and inhomogeneous particle distributions, which result in significant errors with respect to pressure and granular temperature. In order to allow for a fair assessment of different implementations, a novel compound sphere particle shape is devised for comparison to reliable benchmark data generated from systems of spherical particles. The Multi-Sphere Discrete Element Method is utilised to examine two implementations of these conditions. The commonly used Naive approach results in the aforementioned unphysical effects, which are numerical artefacts causing deviations from the benchmark results of up to one order of magnitude. Meanwhile, the proposed consistent implementation fulfils the fundamental requirements of Lees-Edwards boundary conditions and produces data which are in excellent agreement with the benchmark results, as well as the available literature. Comparing the aforementioned implementations, general principles are developed for implementing Lees-Edwards boundary conditions for the Multi-Sphere Discrete Element Method.

Periodic boundary conditions for arbitrary deformations in molecular dynamics simulations

A generalization of the Lees-Edwards periodic boundary conditions (gLE-PBC) for molecular dynamics (MD) simulations is developed to allow for arbitrary deformations to be applied to the domain. The gLE-PBC domain remains a rectangular cuboid regardless of the applied deformation in contrast with the Lagrangian-rhomboid periodic boundary conditions (LR-PBC) where the domain deforms according to the applied deformation. The kinematics of gLE-PBC are validated against pure shear. The gLE-PBC method for interacting systems is then validated against the LR-PBC method and analytical solutions for a solid under isotropic compression, one-dimensional shearing, three-dimensional extension and shearing and for a liquid under Couette flow. Bulk physical properties extracted from the gLE-PBC simulations agree well with values calculated from equilibrium MD simulations. Three dimensional shearing and a deformation with a full velocity gradient matrix are also simulated, showing the range of problems gLE-PBC can explore.

Prediction of shear thickening of particle suspensions in viscoelastic fluids by direct numerical simulation

Abstract

Modified thermal periodic Poiseuille and Lees-Edwards boundary conditions for energy conservative dissipative particle dynamics

Two kinds of thermal boundary conditions without wall particles in energy conservative dissipative particle dynamics method are investigated in present paper. Firstly, the original thermal periodic Poiseuille flow (PPF) was improved to avoid the unreasonable temperature shift by rebuilding the thermal equilibrium. Then a novel thermal Lees-Edwards (LE) boundary condition was proposed. The constant temperature gradient and uniform density distribution are obtained by rescaling the interaction between particle pairs, and the velocity components of particles for particles re-entering the boundaries. Finally, both the temperature and velocity gradient are integrated into the thermal LE to simulate a constant temperature different in a shear flow. The thermal LE provides a new way to study the thermal-mechanical coupled problems with less computational cost and artificial fluctuation of properties compared with traditional wall particle boundary conditions. Furthermore, it can also be easily extended to related particle-based methods, such as molecular dynamics and Brownian dynamics, etc.

Computational study of transient shear banding in soft jammed solids.

We have designed three-dimensional numerical simulations of a soft spheres model, with size polidispersity and in athermal conditions, to study the transient shear banding that occurs during yielding of jammed soft solids. We analyze the effects of different types of drag coefficients used in the simulations and compare the results obtained using Lees-Edwards periodic boundary conditions with the case in which the same model solid is confined between two walls. The specific damping mechanism and the different boundary conditions indeed modify the load curves and the velocity profiles in the transient regime. Nevertheless, we find that the presence of a stress overshoot and of a related transient banding phenomenon, for large enough samples, is a robust feature for overdamped systems, where their presence do not depend on the specific drag used and on the different boundary conditions.

Steady-state rheology and structure of soft hybrid mixtures of liquid crystals and magnetic nanoparticles.

Using non-equilibrium molecular dynamics simulations, we study the rheology of a model hybrid mixture of liquid crystals (LCs) and dipolar soft spheres (DSS) representing magnetic nanoparticles. The bulk isotropic LC-DSS mixture is sheared with different shear rates using Lees-Edwards periodic boundary conditions. The steady-state rheological properties and the effect of the shear on the microstructure of the mixture are studied for different strengths of the dipolar coupling, λ, among the DSS. We find that at large shear rates, the mixture shows a shear-thinning behavior for all considered values of λ. At low and intermediate values of λ, a crossover from Newtonian to non-Newtonian behavior is observed as the rate of applied shear is increased. In contrast, for large values of λ, such a crossover is not observed within the range of shear rates considered. Also, the extent of the non-Newtonian regime increases as λ is increased. These features can be understood via the shear-induced changes of the microstructure. In particular, the LCs display a shear-induced isotropic-to-nematic transition at large shear rates with a shear-rate dependent degree of nematic ordering. The DSS show a shear-induced nematic ordering only for large values of λ, where the particles self-assemble into chains. Moreover, at large λ and low shear rates, our simulations indicate that the DSS form ferromagnetic domains.

Isomorph invariance of dynamics of sheared glassy systems.

We study hidden scale invariance in the glassy phase of the Kob-Andersen binary Lennard-Jones system. After cooling below the glass transition, we generate a so-called isomorph from the fluctuations of potential energy and virial in the NVT ensemble: a set of density, temperature pairs for which structure and dynamics are identical when expressed in appropriate reduced units. To access dynamical features, we shear the system using the SLLOD algorithm coupled with Lees-Edwards boundary conditions and study the statistics of stress fluctuations and the particle displacements transverse to the shearing direction. We find good collapse of the statistical data, showing that isomorph theory works well in this regime. The analysis of stress fluctuations, in particular the distribution of stress changes over a given strain interval, allows us to identify a clear signature of avalanche behavior in the form of an exponential tail on the negative side. This feature is also isomorph invariant. The implications of isomorphs for theories of plasticity are discussed briefly.

Analysis of two partially-saturated-cell methods for lattice Boltzmann simulation of granular suspension rheology

The lattice Boltzmann method (LBM) has been widely used to simulate fluid–solid flows with various approaches to couple the two phases. We study the partially-saturated-cell (PSC) approach proposed by Nobel and Torcyznski [27], which modifies the LB equation by an additional solid collision term weighted by the lattice solid fraction. We analyse two different PSC schemes with regard to its capability of accurately computing the hydrodynamic stresslet, which is essential to computing the stress, hence the rheology of suspensions of granular particles. Through simulations of single and pair particles in a simple shear flow field, we show that a commonly used solid collision term based on non-equilibrium bounce-back fails to correctly capture the stresslet, although can result in a numerically accurate hydrodynamic torque, when compared to the analytical solutions. In contrast, a model using superposition, which has neither hitherto been analysed nor extensively applied, is demonstrated to be able to accurately calculate both the stresslet and the torque. We finally highlight the importance of using the correct model when simulating suspension rheology, showing the viscosity obtained in simulations of hundreds of mono-disperse particles at various solid fractions sheared homogeneously using the Lees-Edwards boundary condition.

Strong convergence of integrators for nonequilibrium Langevin dynamics

ABSTRACTSeveral numerical schemes are proposed for the solution of Nonequilibrium Langevin Dynamics (NELD), and the strong rate of convergence for each scheme is analyzed. The schemes considered here employ specialised periodic boundary conditions that deform with the flow, namely Lees-Edwards and Kraynik-Reinelt boundary conditions and their generalisations. We show that care must be taken when implementing standard stochastic integration schemes with these boundary conditions in order to avoid a breakdown in the strong order of convergence.

Diffusion of size bidisperse spheres in dense granular shear flow.

Diffusion is an important particle behavior in granular flow. Although granular diffusion has been studied for decades, the diffusion of size bidisperse particles has not been well understood. In this paper, discrete element method simulations with the Lees-Edwards boundary condition are performed to quantify the relation between the diffusion coefficient (D) and flow parameters for size bidisperse spheres in dense granular flow. The influences of the shear rate (γ[over ̇]), the solids fraction (f), and the diameter ratio (D_{LS}) of particles on diffusion are studied. The effects of the friction coefficient (μ) and the restitution coefficient (e) are also investigated. The results indicate that while small particles diffuse faster than large particles in a binary system the volume weighted average diffusion coefficient is proportional to the shear rate and the square of the volume weighted average particle diameter, d^{2}, and it is inversely proportional to the solids fraction. The quantified relation is given as D=k_{d}γ[over ̇]d^{2}, where k_{d}=0.0186/f, and this relation is not sensitive to the diameter ratio for D_{LS}≤3. The diffusion coefficient is not sensitive to the friction coefficient except for the extreme condition where μ<0.1, and it is also not sensitive to the restitution coefficient between 0.3 and 0.9.

Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity.

We develop topological methods for characterizing the relationship between polymer chain entanglement and bulk viscoelastic responses. We introduce generalized Linking Number and Writhe characteristics that are applicable to open linear chains. We investigate the rheology of polymeric chains entangled into weaves with varying topologies and levels of chain density. To investigate viscoelastic responses, we perform non-equilibrium molecular simulations over a range of frequencies using sheared Lees–Edwards boundary conditions. We show how our topological characteristics can be used to capture key features of the polymer entanglements related to the viscoelastic responses. We find there is a linear relation over a significant range of frequencies between the mean absolute Writhe and the Loss Tangent . We also find an approximate inverse linear relationship between the mean absolute Periodic Linking Number and the Loss Tangent . Our results show some of the ways topological methods can be used to characterize chain entanglements to better understand the origins of mechanical responses in polymeric materials.

Shear-induced aggregation or disaggregation in edible oils: Models, computer simulation, and USAXS measurements

The effects of shear upon the aggregation of solid objects formed from solid triacylglycerols (TAGs) immersed in liquid TAG oils were modeled using Dissipative Particle Dynamics (DPD) and the predictions compared to experimental data using Ultra-Small Angle X-ray Scattering (USAXS). The solid components were represented by spheres interacting via attractive van der Waals forces and short range repulsive forces. A velocity was applied to the liquid particles nearest to the boundary, and Lees-Edwards boundary conditions were used to transmit this motion to non-boundary layers via dissipative interactions. The shear was created through the dissipative forces acting between liquid particles. Translational diffusion was simulated, and the Stokes-Einstein equation was used to relate DPD length and time scales to SI units for comparison with USAXS results. The SI values depended on how large the spherical particles were (250 nm vs. 25 nm). Aggregation was studied by (a) computing the Structure Function and (b) quantifying the number of pairs of solid spheres formed. Solid aggregation was found to be enhanced by low shear rates. As the shear rate was increased, a transition shear region was manifested in which aggregation was inhibited and shear banding was observed. Aggregation was inhibited, and eventually eliminated, by further increases in the shear rate. The magnitude of the transition region shear, γ̇t, depended on the size of the solid particles, which was confirmed experimentally.

Viscometric flow for a many-body dissipative particle dynamics (MDPD) fluid with Lees–Edwards boundary condition

Viscometric properties of polymer are explored by the many-body dissipative particle dynamics (MDPD) using Lees–Edwards boundary conditions. The equation of state for the MDPD system is modified by fitting the density correction to different values of the cut-off radius. Due to the many-body interactions in MDPD, the viscosity contributed from the conservative force increases considerably with increasing repulsive coefficient, density and cut-off radius, and cannot be ignored compared to the ‘standard’ DPD case. The influence of these parameters on the MDPD viscosity is investigated, and we propose an equation to predict the viscosity in MDPD model. Additionally, the dumbbell polymer suspension model is investigated in the MDPD fluid, and the relations concerning first normal stress difference and shear rate, the relaxation time and spring constant, are consistent to theoretical works. We conclude that the MDPD model can be used to investigate the dynamics of non-Newtonian droplets.

Pairwise adaptive thermostats for improved accuracy and stability in dissipative particle dynamics

We examine the formulation and numerical treatment of dissipative particle dynamics (DPD) and momentum-conserving molecular dynamics. We show that it is possible to improve both the accuracy and the stability of DPD by employing a pairwise adaptive Langevin thermostat that precisely matches the dynamical characteristics of DPD simulations (e.g., autocorrelation functions) while automatically correcting thermodynamic averages using a negative feedback loop. In the low friction regime, it is possible to replace DPD by a simpler momentum-conserving variant of the Nosé–Hoover–Langevin method based on thermostatting only pairwise interactions; we show that this method has an extra order of accuracy for an important class of observables (a superconvergence result), while also allowing larger timesteps than alternatives. All the methods mentioned in the article are easily implemented. Numerical experiments are performed in both equilibrium and nonequilibrium settings; using Lees–Edwards boundary conditions to induce shear flow.

Rheological evaluation of colloidal dispersions using the smoothed profile method: formulation and applications

The smoothed profile method is extended to study the rheological behaviour of colloidal dispersions under shear flow by using the Lees–Edwards boundary conditions. We start with a reformulation of the smoothed profile method, a direct numerical simulation method for colloidal dispersions, so that it can be used with the Lees–Edwards boundary condition, under steady or oscillatory-shear flow. By this reformulation, all the resultant physical quantities, including local and total shear stresses, become available through direct calculation. Three simple rheological simulations are then performed for (1) a spherical particle, (2) a rigid bead chain and (3) a collision of two spherical particles under shear flow. Quantitative validity of these simulations is examined by comparing the viscosity with that obtained from theory and Stokesian dynamics calculations. Finally, we consider the shear-thinning behaviour of concentrated colloidal dispersions.

Anisotropy and probe-medium interactions in the microrheology of nematic fluids

A theoretical formalism is presented to analyze and interpret microrheology experiments in anisotropic fluids with nematic order. The predictions of that approach are examined in the context of a simple coarse-grained molecular model which is simulated using nonequilibrium molecular dynamics calculations. The proposed formalism is used to study the effect of confinement, the type of anchoring at the probe-particle surface, and the strength of the nematic field on the rheological response functions obtained from probe-particle active microrheology. As expected, a stronger nematic field leads to increased anisotropy in the rheological response of the material. It is also found that the defect structures that arise around the probe particle, which are determined by the type of anchoring and the particle size, have a significant effect on the rheological response observed in microrheology simulations. Independent estimates of the bulk dynamic modulus of the model nematic fluid considered here are obtained from small-amplitude oscillatory shear simulations with Lees–Edwards boundary conditions. The results of simulations indicate that the dynamic modulus extracted from particle-probe microrheology is different from that obtained in the absence of the particle, but that the differences decrease as the size of the defect also decreases. Importantly, the results of the nematic microrheology theory proposed here are in much closer agreement with simulations than those from earlier formalisms conceived for isotropic fluids. As such, it is anticipated that the theoretical framework advanced in this study could provide a useful tool for interpretation of microrheology experiments in systems such as liquid crystals and confined macromolecular solutions or gels.

Thermostatic and rheological responses of DPD fluid to extreme shear under modified Lees-Edwards boundary condition.

Thermodynamic, hydrodynamic and rheological interactions between velocity-dependent thermostats of Lowe-Andersen (LA) and Nosé-Hoover-Lowe-Andersen (NHLA), and modified Lees-Edwards (M-LEC) boundary condition were studied in the context of Dissipative Particle Dynamics method. Comparisons were made with original Lees-Edwards method to characterise the improvements that M-LEC offers in conserving the induced shear momentum. Different imposed shear velocities, heat bath collision/exchange frequencies and thermostating probabilities were considered. The presented analyses addressed an unusual discontinuity in momentum transfer that appeared in form of nonphysical jumps in velocity and temperature profiles. The usefulness of M-LEC was then quantified by evaluating the enhancements in obtained effective shear velocity, effective shear rate, Péclet number, and dynamic viscosity. System exchange frequency (Γ) with Maxwellian heat bath was found to play an important role, in that its larger values facilitated achieving higher shear rates with proper temperature control at the cost of deviation from an ideal momentum transfer. Similar dynamic viscosities were obtained under both shearing modes between LA and NHLA thermostats up to Γ = 10, whilst about twice the range of viscosity (1 < η < 20) was calculated for M-LEC at larger probabilities (ΓΔt > %). The main benefits of this modification were to facilitate momentum flow from shear boundaries to the system bulk. In addition, it was found that there exist upper thresholds for imposing shear on the system beyond which temperature cannot be controlled properly and nonphysical jumps reappear.