Periodic Arrays Of Spheres Research Articles

Robust and non-robust bound states in the continuum in rotationally symmetric periodic waveguides.

A fiber grating and a one-dimensional (1D) periodic array of spheres are examples of rotationally symmetric periodic (RSP) waveguides. It is well known that bound states in the continuum (BICs) may exist in lossless dielectric RSP waveguides. Any guided mode in an RSP waveguide is characterized by an azimuthal index m, the frequency ω, and Bloch wavenumber β. A BIC is a guided mode, but for the same m, ω and β, cylindrical waves can propagate to or from infinity in the surrounding homogeneous medium. In this paper, we investigate the robustness of nondegenerate BICs in lossless dielectric RSP waveguides. The question is whether a BIC in an RSP waveguide with a reflection symmetry along its axis z, can continue its existence when the waveguide is perturbed by small but arbitrary structural perturbations that preserve the periodicity and the reflection symmetry in z. It is shown that for m = 0 and m ≠ 0, generic BICs with only a single propagating diffraction order are robust and non-robust, respectively, and a non-robust BIC with m ≠ 0 can continue to exist if the perturbation contains one tunable parameter. The theory is established by proving the existence of a BIC in the perturbed structure mathematically, where the perturbation is small but arbitrary, and contains an extra tunable parameter for the case of m ≠ 0. The theory is validated by numerical examples for propagating BICs with m ≠ 0 and β ≠ 0 in fiber gratings and 1D arrays of circular disks.

How Stickiness Can Speed Up Diffusion in Confined Systems.

The paradigmatic model for heterogeneous media used in diffusion studies is built from reflecting obstacles and surfaces. It is well known that the crowding effect produced by these reflecting surfaces slows the dispersion of Brownian tracers. Here, using a general adsorption desorption model with surface diffusion, we show analytically that making surfaces or obstacles attractive can accelerate dispersion. In particular, we show that this enhancement of diffusion can exist even when the surface diffusion constant is smaller than that in the bulk. Even more remarkably, this enhancement effect occurs when the effective diffusion constant, when restricted to surfaces only, is lower than the effective diffusivity with purely reflecting boundaries. We give analytical formulas for this intriguing effect in periodic arrays of spheres as well as undulating microchannels. Our results are confirmed by numerical calculations and MonteCarlo simulations.

Microrheology analysis in molecular dynamics simulations: Finite box size correction

In single-particle microrheology, the viscoelastic properties of a complex fluid can be extracted using the generalized Stokes–Einstein (GSE) relation by embedding a micrometer-sized particle and tracking its motion through the fluid. Applying the same analysis to molecular dynamics simulations can result in overestimated G∗ values because of the hydrodynamic interaction between the probe bead and its periodic images. We derive a simple correction to the GSE equation by implementing an analytical solution for Stokes drag on a periodic array of spheres, which allows smaller box sizes to be simulated while still retaining accuracy. Fluid and particle inertia are neglected, although the approach used here might be generalized to include them. The correction is applied to molecular dynamics simulations of a coarse-grained polymer melt. For several bead-to-box-size ratios R/L and two probe sizes, we measure the mean squared displacement and calculate G∗ using both the original GSE and the corrected hydrodynamic Stokes–Einstein (HSE) equations. These results are compared to small amplitude oscillatory non-equilibrium molecular dynamics (NEMD) simulations, which show that the HSE analysis correctly predicts the G∗ values at low frequencies but breaks down when either fluid inertia is important or the bead size is too small to “see” a continuum. For small R/L, the GSE and HSE results are similar, although a small correction is still required for the finite box size and computational cost is significantly larger. The HSE equation allows the use of smaller box sizes, reducing computational costs by more than an order of magnitude.

Hydrodynamic effects on the aggregation of nanoparticles in porous media

The aggregation of spherical nanoparticles as they propagate through porous media is explored using lattice Boltzmann simulations and tracking of the trajectoties of individual particles. The porous media are modeled as periodic arrays of spheres in different packing configurations. The effects of interparticle interactions on particle aggregation are treated through a single aggregation probability upon particle collision. Fast aggregation is represented by an aggregation probability with values close to one, and slow aggregation is represented by lower aggregation probability. An algorithm that accounts for the different time scales between hydrodynamics and Brownian motion is devised and validated. It is found that primary particle size, initial particle concentration, injection flow rate, and aggregation probability significantly impact the dynamics of the aggregation. Calculations of the transient mean size of the aggregates show that the aggregation rate is high near the entrance of the porous medium, yet dramatically decreases when moving farther downstream. The reason is that the growth of larger aggregates slows the aggregation process. It is also found that the bigger aggregates are formed in the pore space near the pore matrix surface, where particle residence time is long enough to allow the aggregates to grow. An empirical correlation based on measurable parameters is proposed for the prediction of the mean aggregate size.

Apparent permeability of flow through periodic arrays of spheres with first-order slip

A lattice Boltzmann bounce-back scheme that can simulate the first-order slip boundary condition at a fluid-solid interface is derived and verified with analytical and asymptotic solutions for shear and pressure driven flows in channels, through square arrays of cylinders, and in a Couette cell. We then simulated Stokes flows with surface slip through simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) arrays of spheres and established the slip (apparent) permeability of these arrays as a function of the no-slip (absolute) permeability, the slip length and the porosity. It is found that ks/kns, the ratio between the slip permeability and the no-slip permeability, increases with increasing slip length and increases with decreasing porosity owning to enhancement of flow through the gaps between particles. Interestingly, ks/kns is nearly independent of sphere configuration when the dimensionless slip length is small; with increasing dimensionless slip length, however, different sphere configurations start to have different ks/kns. The dependence of ks/kns on the slip length is slightly nonlinear owning to the curvature of the surface and this dependence can be correlated by quadratic equations.

Electromagnetic coupling and array packing induce exchange of dominance on complex modes in 3D periodic arrays of spheres with large permittivity

We investigate the effect on wave propagation of array packing and electromagnetic coupling between spheres in a three-dimensional (3D) lattice of microspheres with large permittivity that exhibit strong magnetic polarizability. We report on the complex wavenumber of Bloch waves in the lattice when each sphere is assumed to possess both electric and magnetic dipoles and full electromagnetic coupling is accounted for. While for small material-filling fractions we always determine one dominant mode with low attenuation constant, the same does not happen for large filling fractions, when electromagnetic coupling is included. In the latter case we peculiarly observe two dominant modes with low attenuation constant, dominant in different frequency ranges. The filling fraction threshold for which two dominant modes appear varies for different metamaterial constituents, as proven by considering spheres made by either titanium dioxide or lead telluride. As further confirmation of our findings, we retrieve the complex propagation constant of the dominant mode(s) via a field fitting procedure employing two sets of waves (direct and reflected) pertaining to two distinct modes, strengthening the presence of the two distinct dominant modes for increasing filling fractions. However, given that one mode only, with transverse polarization, at any given frequency, is dominant and able to propagate inside the lattice, we are able to accurately treat the metamaterial that is known to exhibit artificial magnetism as a homogeneous material with effective parameters, such as the refractive index. Results clearly show that the account of both electric and magnetic scattering processes in evaluating all electromagnetic intersphere couplings is essential for a proper description of the electromagnetic propagation in lattices.

The raspberry model for hydrodynamic interactions revisited. I. Periodic arrays of spheres and dumbbells.

The so-called "raspberry" model refers to the hybrid lattice-Boltzmann and Langevin molecular dynamics scheme for simulating the dynamics of suspensions of colloidal particles, originally developed by Lobaskin and Dünweg [New J. Phys. 6, 54 (2004)], wherein discrete surface points are used to achieve fluid-particle coupling. This technique has been used in many simulation studies on the behavior of colloids. However, there are fundamental questions with regards to the use of this model. In this paper, we examine the accuracy with which the raspberry method is able to reproduce Stokes-level hydrodynamic interactions when compared to analytic expressions for solid spheres in simple-cubic crystals. To this end, we consider the quality of numerical experiments that are traditionally used to establish these properties and we discuss their shortcomings. We show that there is a discrepancy between the translational and rotational mobility reproduced by the simple raspberry model and present a way to numerically remedy this problem by adding internal coupling points. Finally, we examine a non-convex shape, namely, a colloidal dumbbell, and show that the filled raspberry model replicates the desired hydrodynamic behavior in bulk for this more complicated shape. Our investigation is continued in de Graaf et al. [J. Chem. Phys. 143, 084108 (2015)], wherein we consider the raspberry model in the confining geometry of two parallel plates.

Periodic homogenization and consistent estimates of transport parameters through sphere and polyhedron packings in the whole porosity range

This paper presents a study of transport parameters (diffusion, dynamic permeability, thermal permeability, trapping constant) of porous media by combining the homogenization of periodic media (HPM) and the self-consistent scheme (SCM) based on a bicomposite spherical pattern. The link between the HPM and SCM approaches is first established by using a systematic argument independent of the problem under consideration. It is shown that the periodicity condition can be replaced by zero flux and energy through the whole surface of the representative elementary volume. Consequently the SCM solution can be considered as a geometrical approximation of the local problem derived through HPM for materials such that the morphology of the period is "close" to the SCM pattern. These results are then applied to derive the estimates of the effective diffusion, the dynamic permeability, the thermal permeability and the trapping constant of porous media. These SCM estimates are compared with numerical HPM results obtained on periodic arrays of spheres and polyhedrons. It is shown that SCM estimates provide good analytical approximations of the effective parameters for periodic packings of spheres at porosities larger than 0.6, while the agreement is excellent for periodic packings of polyhedrons in the whole range of porosity.

Analysis of unsteady forces in ordered arrays of monodisperse spheres

(Received 19 July 2004 and in revised form 12 October 2005) Time-dependent flows of a Newtonian fluid through periodic arrays of spheres were simulated using the lattice-Boltzmann scheme. By applying a constant body force per unit mass to the fluid, a steady background fluid flow through the array of stationary spheres was first established. Subsequently, a small-amplitude perturbation to the body force, which varied periodically in time, was added and the long-time behaviour of the unsteady flow fields and the forces on the particles were determined. From the simulations, the pressure and friction (shear) forces acting on the particles were determined for a range of conditions. Results on simple cubic lattices are presented. Computations spanned a range of particle volume fractions (0.1 <φ< 0.4), background flow Reynolds numbers (0.25 Rep 60, where Rep =2 auf /ν )a nd oscillatory flow Reynolds numbers (0.9 Reω 420 with Reω =2 a 2 ω/ν). Here uf is the superficial velocity of the fluid through the bed, a is the particle radius, ν is the kinematic viscosity of the fluid, and ω is the oscillation frequency. In the limit of Reω → 0 the quasi-steady-state drag force was obtained. At low Rep this force approached the steady-state drag force, while its increase with Rep was stronger than the steady-state drag force, similar to that for isolated spheres given by Mei et al. (J. Fluid Mech., vol. 233, 1991, p. 613). The unsteady force was decomposed into pressure and friction components. The phase angles of these components in the limit Reω →∞ indicate that the virtual mass force contributes to the unsteady pressure force while the history force contributes to the friction force. The remainder of the unsteady friction and pressure forces is attributed to unsteady drag force. The apparent virtual mass coefficient was found to vary from ∼0.5 at high Reω, which is the well-known limit for isolated spheres in inviscid flows, to ∼1. 0a t low Reω. This change is clearly a consequence of viscous effects. The Reω at which the transition between these limits occurs increases with φ. The history force exhibits a strong decay towards lower values of Reω in accordance with the results of Mei et al. (1991) for isolated spheres; however, the Reω value at which this decay sets in increases appreciably with φ .T hisφ-dependence is associated with the limited separation between the particles available for the Stokes boundary layer. It was found that the unsteady drag coefficient β � varies with Reω .A t lowRep ,t he drag coefficient initially decreases with increasing Reω, passes through a minimum and then increases strongly. With increasing Reω the relative contribution of pressure and friction forces to the unsteady drag force changes.

Multireflection boundary conditions for lattice Boltzmann models.

We present a general framework for several previously introduced boundary conditions for lattice Boltzmann models, such as the bounce-back rule and the linear and quadratic interpolations. The objectives are twofold: first to give theoretical tools to study the existing link-type boundary conditions and their corresponding accuracy; second to design boundary conditions for general flows which are third-order kinetic accurate. Using these new boundary conditions, Couette and Poiseuille flows are exact solutions of the lattice Boltzmann models for a Reynolds number Re=0 (Stokes limit) for arbitrary inclination with the lattice directions. Numerical comparisons are given for Stokes flows in periodic arrays of spheres and cylinders, linear periodic array of cylinders between moving plates, and for Navier-Stokes flows in periodic arrays of cylinders for Re<200. These results show a significant improvement of the overall accuracy when using the linear interpolations instead of the bounce-back reflection (up to an order of magnitude on the hydrodynamics fields). Further improvement is achieved with the new multireflection boundary conditions, reaching a level of accuracy close to the quasianalytical reference solutions, even for rather modest grid resolutions and few points in the narrowest channels. More important, the pressure and velocity fields in the vicinity of the obstacles are much smoother with multireflection than with the other boundary conditions. Finally the good stability of these schemes is highlighted by some simulations of moving obstacles: a cylinder between flat walls and a sphere in a cylinder.

The hydrodynamic force and torque on a bounded sphere in Poiseuille flow

The lattice Boltzmann (LB) method is used to study the hydrodynamic force and torque acting on a sphere held stationary between parallel plates in pressure-driven flow. This and associated flow configurations are explored in this paper. LB results are in excellent agreement with existing theory and numerical results for simple pressure-driven flow between parallel plates, for flow through a periodic medium of spheres [Zick AA, Homsy GM. Stokes flow through periodic arrays of spheres. Journal of Fluid Mechanics 1982; 115: 13], and for the force and torque acting on a sphere held fixed at the quarter vertical position in a pressure-driven flow between parallel plates. In the latter case, LB calculations reveal a screening effect caused by neighboring periodic images of the test sphere. It is shown that the test sphere is hydrodynamically decoupled from its periodic images when separated by approximately 30 sphere radii. LB results for force and torque as a function of sphere height and flow cell height are also reported. Copyright © 2001 John Wiley & Sons, Ltd.

The hydrodynamic force and torque on a bounded sphere in Poiseuille flow

The lattice Boltzmann (LB) method is used to study the hydrodynamic force and torque acting on a sphere held stationary between parallel plates in pressure-driven flow. This and associated flow configurations are explored in this paper. LB results are in excellent agreement with existing theory and numerical results for simple pressure-driven flow between parallel plates, for flow through a periodic medium of spheres [Zick AA, Homsy GM. Stokes flow through periodic arrays of spheres. Journal of Fluid Mechanics 1982; 115: 13], and for the force and torque acting on a sphere held fixed at the quarter vertical position in a pressure-driven flow between parallel plates. In the latter case, LB calculations reveal a screening effect caused by neighboring periodic images of the test sphere. It is shown that the test sphere is hydrodynamically decoupled from its periodic images when separated by approximately 30 sphere radii. LB results for force and torque as a function of sphere height and flow cell height are also reported. Copyright © 2001 John Wiley & Sons, Ltd.

Ultrasonic diffraction by a square periodic array of spheres

Shear waves propagated as a result of the diffraction of a broadband normally incident plane ultrasonic longitudinal pulse by a square periodic array of spherical elastic inclusions are measured at specific angles from the normal. As expected, peaks in the spectra were observed at frequencies given by the condition of constructive interference.

Resolution Effects in Dissipative Particle Dynamics Simulations

Dissipative Particle Dynamics (DPD) simulations were performed to investigate resolution or "coarse graining" effects on the simulation results. Fluid flow through a periodic array of spheres has been studied as a model for fluid filtration into a porous medium. In our model system, it appears that quantitatively correct results for the dimensionless drag can be obtained for relatively small system sizes. For higher solid volume fractions, it is necessary to increase the system size to avoid finite size and resolution effects. Simulations of colloidal spheres suspended in a DPD fluid show effective attraction between the large colloid particles, causing depletion aggregation. This effect may be expected as a consequence of the coarse-grained nature of the DPD fluid. By imposing a steady shear rate the aggregation can be suppressed. The results show that for dilute suspensions, the Brownian noise in the particle interactions causes an effective colloid polydispersity, which suppresses aggregation effects.

Electrical Double-Layer Interactions of Regular Arrays of Spheres

Electrostatic interactions are computed for three-dimensional regular arrays of spheres. Tetrahedral array of four spheres, octahedral array of six spheres, cubic array of eight spheres, and periodic arrays of spheres in body-centered cubic (BCC) and face-centered cubic (FCC) structures are considered. Based on the linearized Poisson-Boltzmann equation the electrostatic free energy is computed for both constant surface potential and constant surface charge conditions. Facilitating the complex geometry of the problem, the boundary element method is used. Test results for two-sphere interactions are in excellent agreement with existing exact solutions even at very small Debye length. The results for multisphere interactions are compared with those obtained by the pairwise additivity approximation. For BCC and FCC arrays comparison is made to the results from the pairwise additivity approximation and from the spherical cell model. At small Debye length the pairwise additivity approximation predicts the interactions reasonably well. At large Debye length, however, the cell model furnishes much better estimates. In addition, the electrostatic free energy difference between BCC and FCC structures at given particle concentration is found to be too small to assess any physical significance.

Electromagnetic interactions with rough metal surfaces

When surfaces are structured on the scale of the wavelength, we can expect incident light to be strongly modified by the surface. This is especially the case when the surface is metallic. We have developed a formalism for computing these modifications, closely analogous to electron scattering theory, which we briefly review and present some results for optical properties of, and electron energy loss in, colloids. Our main theme is another effect associate with rough or structured metallic surfaces: Surface Enhanced Raman Scattering, or SERS. We model the rough surface by a periodic array of spheres and obtain the correct magnitude for the enhancement and for the frequency shifts observed.

Harmonic stokes flow through periodic porous media: a 3D boundary element method

Our interest is in dynamic filtration through periodic, porous, saturated media. More precisely, here we develop a three-dimensional numerical model, based on boundary element methods, to compute the dynamic permeability over a wide range of such media. This generalized Darcy coefficient is obtained by the homogenization process applied to a periodic, deformable, porous medium under dynamic solicitations. An unusual choice of Green functions is made. A simple numerical procedure is used for the treatment of the periodic boundary conditions. Recent advances to treat singular integrals are employed and extended to our case. The method is tested on simple examples where theoretical results are available. In the static case results are compared with many previous results on periodic arrays of spheres. New results are given in the dynamic case. The scaling behavior for dynamic permeability in porous media is checked and discussed.

Conductivity of incompletely filled periodic arrays of spheres

We consider the problem of determining the effective thermal conductivity k ' of a composite that consists of spherical particles of thermal conductivity αk arranged on the lattice of a periodic array and surrounded by a matrix of conductivity k when a fraction f of the lattice sites is unoccupied by the particles. This may be regarded as a model of a composite medium in which the dispersed phase consists of relatively large aggregates. A method is developed to account for the long range multiparticle interactions and to determine k ' powers of a , where a is the ratio of the particle diameter to the lattice size. By using this method we determine k ' correct to O ( a 14 ) for all values of f and α . We also determine the ratio of effective conductivities k ' / k * , where k * is the conductivity at f = 0, for the case of when the particles are highly conducting ( α ⪢1) and are nearly touching their neighbours ( a → 1) via Monte-Carlo simulations including up to 3 x 10 4 particles. We then combine the two results to obtain estimates of k ' that are expected to be accurate for a wide range of α , a , and f .

ADXPS/STEM studies of surface and bulk microphase behavior in block- and graft-copolymers and their blends

This study of surface and bulk microphase separation compares different block chemistries and topologies. The molecular weight and percentage of the siloxane block dominated the phase behavior, which showed coincident transitions at surfaces and in the bulk. Conditions were found where periodic arrays of spheres, lamellae or cylinders formed in the bulk, and spheres were observed at some of the surfaces. Thus quantitative interpretation of ADXPS depth profiles is questionable. Blends of copolymers with homopolymer show pronounced surface segregation with a critical concentration at ≈1 wt% siloxane in the bulk that saturates the surface. The extent of surface and bulk phase separation is strongly effected by the kinetics of solvent evaporation or thermoforming.

Slow flow through a periodic array of spheres

Slow flow through a periodic array of spheres is studied theoretically, and the drag force by the fluid on a sphere forming the periodic array is calculated using a modification of the method developed by Hashimoto (1959). Results for the complete range of volume fraction c of spheres are given for simple cubic, body-centered cubic, and face-centered cubic arrays and these agree well with the corresponding values reported by previous investigators. Also, series expansions for the drag force to 0( c 10 ) are derived for each of these cubic arrays. The method is also applied to determine the drag force to 0( c 3 ) on infinitely long cylinders in square and hexagonal arrays.